| L(s) = 1 | + 7·2-s + 28·4-s + 7·5-s + 7-s + 84·8-s + 49·10-s − 11-s + 7·14-s + 210·16-s + 9·17-s + 9·19-s + 196·20-s − 7·22-s + 8·23-s + 28·25-s + 28·28-s + 4·29-s + 16·31-s + 462·32-s + 63·34-s + 7·35-s + 2·37-s + 63·38-s + 588·40-s + 41-s − 10·43-s − 28·44-s + ⋯ |
| L(s) = 1 | + 4.94·2-s + 14·4-s + 3.13·5-s + 0.377·7-s + 29.6·8-s + 15.4·10-s − 0.301·11-s + 1.87·14-s + 52.5·16-s + 2.18·17-s + 2.06·19-s + 43.8·20-s − 1.49·22-s + 1.66·23-s + 28/5·25-s + 5.29·28-s + 0.742·29-s + 2.87·31-s + 81.6·32-s + 10.8·34-s + 1.18·35-s + 0.328·37-s + 10.2·38-s + 92.9·40-s + 0.156·41-s − 1.52·43-s − 4.22·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{7} \cdot 3^{14} \cdot 5^{7} \cdot 89^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{7} \cdot 3^{14} \cdot 5^{7} \cdot 89^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9590.981366\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9590.981366\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - T )^{7} \) |
| 3 | \( 1 \) |
| 5 | \( ( 1 - T )^{7} \) |
| 89 | \( ( 1 + T )^{7} \) |
| good | 7 | \( 1 - T + 12 T^{2} + 4 T^{3} + 20 T^{4} + 27 T^{5} - 23 T^{6} - 556 T^{7} - 23 p T^{8} + 27 p^{2} T^{9} + 20 p^{3} T^{10} + 4 p^{4} T^{11} + 12 p^{5} T^{12} - p^{6} T^{13} + p^{7} T^{14} \) |
| 11 | \( 1 + T + 12 T^{2} + 20 T^{3} + 86 T^{4} + p T^{5} + 351 T^{6} - 4240 T^{7} + 351 p T^{8} + p^{3} T^{9} + 86 p^{3} T^{10} + 20 p^{4} T^{11} + 12 p^{5} T^{12} + p^{6} T^{13} + p^{7} T^{14} \) |
| 13 | \( 1 + 5 T^{2} + 18 T^{3} + 323 T^{4} - 632 T^{5} + 2903 T^{6} + 6572 T^{7} + 2903 p T^{8} - 632 p^{2} T^{9} + 323 p^{3} T^{10} + 18 p^{4} T^{11} + 5 p^{5} T^{12} + p^{7} T^{14} \) |
| 17 | \( 1 - 9 T + 66 T^{2} - 302 T^{3} + 1544 T^{4} - 6931 T^{5} + 38229 T^{6} - 158924 T^{7} + 38229 p T^{8} - 6931 p^{2} T^{9} + 1544 p^{3} T^{10} - 302 p^{4} T^{11} + 66 p^{5} T^{12} - 9 p^{6} T^{13} + p^{7} T^{14} \) |
| 19 | \( 1 - 9 T + 70 T^{2} - 264 T^{3} + 1296 T^{4} - 5139 T^{5} + 1941 p T^{6} - 156264 T^{7} + 1941 p^{2} T^{8} - 5139 p^{2} T^{9} + 1296 p^{3} T^{10} - 264 p^{4} T^{11} + 70 p^{5} T^{12} - 9 p^{6} T^{13} + p^{7} T^{14} \) |
| 23 | \( 1 - 8 T + 121 T^{2} - 618 T^{3} + 5493 T^{4} - 19104 T^{5} + 145101 T^{6} - 419356 T^{7} + 145101 p T^{8} - 19104 p^{2} T^{9} + 5493 p^{3} T^{10} - 618 p^{4} T^{11} + 121 p^{5} T^{12} - 8 p^{6} T^{13} + p^{7} T^{14} \) |
| 29 | \( 1 - 4 T + 55 T^{2} - 2 p T^{3} + 2177 T^{4} - 1884 T^{5} + 88399 T^{6} - 124012 T^{7} + 88399 p T^{8} - 1884 p^{2} T^{9} + 2177 p^{3} T^{10} - 2 p^{5} T^{11} + 55 p^{5} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} \) |
| 31 | \( 1 - 16 T + 187 T^{2} - 1254 T^{3} + 6411 T^{4} - 12336 T^{5} - 38991 T^{6} + 629580 T^{7} - 38991 p T^{8} - 12336 p^{2} T^{9} + 6411 p^{3} T^{10} - 1254 p^{4} T^{11} + 187 p^{5} T^{12} - 16 p^{6} T^{13} + p^{7} T^{14} \) |
| 37 | \( 1 - 2 T + 105 T^{2} - 102 T^{3} + 6591 T^{4} + 218 T^{5} + 287463 T^{6} + 114460 T^{7} + 287463 p T^{8} + 218 p^{2} T^{9} + 6591 p^{3} T^{10} - 102 p^{4} T^{11} + 105 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \) |
| 41 | \( 1 - T + 90 T^{2} - 716 T^{3} + 108 p T^{4} - 47511 T^{5} + 357281 T^{6} - 1611288 T^{7} + 357281 p T^{8} - 47511 p^{2} T^{9} + 108 p^{4} T^{10} - 716 p^{4} T^{11} + 90 p^{5} T^{12} - p^{6} T^{13} + p^{7} T^{14} \) |
| 43 | \( 1 + 10 T + 169 T^{2} + 1380 T^{3} + 14289 T^{4} + 103718 T^{5} + 875313 T^{6} + 5394360 T^{7} + 875313 p T^{8} + 103718 p^{2} T^{9} + 14289 p^{3} T^{10} + 1380 p^{4} T^{11} + 169 p^{5} T^{12} + 10 p^{6} T^{13} + p^{7} T^{14} \) |
| 47 | \( 1 - 13 T + 266 T^{2} - 2976 T^{3} + 34044 T^{4} - 308291 T^{5} + 2552771 T^{6} - 395040 p T^{7} + 2552771 p T^{8} - 308291 p^{2} T^{9} + 34044 p^{3} T^{10} - 2976 p^{4} T^{11} + 266 p^{5} T^{12} - 13 p^{6} T^{13} + p^{7} T^{14} \) |
| 53 | \( 1 - 24 T + 391 T^{2} - 3552 T^{3} + 20129 T^{4} + 29720 T^{5} - 1458865 T^{6} + 15820096 T^{7} - 1458865 p T^{8} + 29720 p^{2} T^{9} + 20129 p^{3} T^{10} - 3552 p^{4} T^{11} + 391 p^{5} T^{12} - 24 p^{6} T^{13} + p^{7} T^{14} \) |
| 59 | \( 1 - 6 T + 181 T^{2} - 1136 T^{3} + 18981 T^{4} - 113562 T^{5} + 1441801 T^{6} - 7447616 T^{7} + 1441801 p T^{8} - 113562 p^{2} T^{9} + 18981 p^{3} T^{10} - 1136 p^{4} T^{11} + 181 p^{5} T^{12} - 6 p^{6} T^{13} + p^{7} T^{14} \) |
| 61 | \( 1 - 23 T + 418 T^{2} - 5120 T^{3} + 55750 T^{4} - 8311 p T^{5} + 4456115 T^{6} - 34993852 T^{7} + 4456115 p T^{8} - 8311 p^{3} T^{9} + 55750 p^{3} T^{10} - 5120 p^{4} T^{11} + 418 p^{5} T^{12} - 23 p^{6} T^{13} + p^{7} T^{14} \) |
| 67 | \( 1 - 5 T + 340 T^{2} - 1194 T^{3} + 50598 T^{4} - 123159 T^{5} + 4654255 T^{6} - 131276 p T^{7} + 4654255 p T^{8} - 123159 p^{2} T^{9} + 50598 p^{3} T^{10} - 1194 p^{4} T^{11} + 340 p^{5} T^{12} - 5 p^{6} T^{13} + p^{7} T^{14} \) |
| 71 | \( 1 - 8 T + 253 T^{2} - 264 T^{3} + 19017 T^{4} + 137160 T^{5} + 927165 T^{6} + 15313040 T^{7} + 927165 p T^{8} + 137160 p^{2} T^{9} + 19017 p^{3} T^{10} - 264 p^{4} T^{11} + 253 p^{5} T^{12} - 8 p^{6} T^{13} + p^{7} T^{14} \) |
| 73 | \( 1 + 2 T + 323 T^{2} + 312 T^{3} + 52097 T^{4} + 19694 T^{5} + 5447211 T^{6} + 1128432 T^{7} + 5447211 p T^{8} + 19694 p^{2} T^{9} + 52097 p^{3} T^{10} + 312 p^{4} T^{11} + 323 p^{5} T^{12} + 2 p^{6} T^{13} + p^{7} T^{14} \) |
| 79 | \( 1 - 7 T + 324 T^{2} - 2856 T^{3} + 53810 T^{4} - 509833 T^{5} + 5883519 T^{6} - 52020304 T^{7} + 5883519 p T^{8} - 509833 p^{2} T^{9} + 53810 p^{3} T^{10} - 2856 p^{4} T^{11} + 324 p^{5} T^{12} - 7 p^{6} T^{13} + p^{7} T^{14} \) |
| 83 | \( 1 - 7 T + 388 T^{2} - 2706 T^{3} + 72398 T^{4} - 491025 T^{5} + 8673015 T^{6} - 52224988 T^{7} + 8673015 p T^{8} - 491025 p^{2} T^{9} + 72398 p^{3} T^{10} - 2706 p^{4} T^{11} + 388 p^{5} T^{12} - 7 p^{6} T^{13} + p^{7} T^{14} \) |
| 97 | \( 1 - 30 T + 795 T^{2} - 13212 T^{3} + 203209 T^{4} - 2397442 T^{5} + 27897435 T^{6} - 271177608 T^{7} + 27897435 p T^{8} - 2397442 p^{2} T^{9} + 203209 p^{3} T^{10} - 13212 p^{4} T^{11} + 795 p^{5} T^{12} - 30 p^{6} T^{13} + p^{7} T^{14} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{14} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.45292607626597335565573062705, −3.43860406587292545635236287137, −3.35923656130842035513406437440, −3.29121971526582136768533121244, −3.08110791027292477602841052541, −2.95059962011580927652285129069, −2.86752211804384052191616813361, −2.62202890379762495104005330835, −2.61629960595294969443985449075, −2.61206074849167186731584996106, −2.55745522874089903424839702634, −2.29727077567748943852450214580, −2.22680908855548882128656882045, −1.86384475435534217338242523915, −1.85722891402777478020609166049, −1.83435980695348845395144148020, −1.78092602462823965700073887785, −1.75301294552548741362429462637, −1.12691303066033277045257198120, −1.06090653682844516953433955298, −1.05244374222143634470201423713, −0.940227278851441558896400361175, −0.919944160814144537640559896009, −0.71284860572115873517318974138, −0.56216829394299677415453076831,
0.56216829394299677415453076831, 0.71284860572115873517318974138, 0.919944160814144537640559896009, 0.940227278851441558896400361175, 1.05244374222143634470201423713, 1.06090653682844516953433955298, 1.12691303066033277045257198120, 1.75301294552548741362429462637, 1.78092602462823965700073887785, 1.83435980695348845395144148020, 1.85722891402777478020609166049, 1.86384475435534217338242523915, 2.22680908855548882128656882045, 2.29727077567748943852450214580, 2.55745522874089903424839702634, 2.61206074849167186731584996106, 2.61629960595294969443985449075, 2.62202890379762495104005330835, 2.86752211804384052191616813361, 2.95059962011580927652285129069, 3.08110791027292477602841052541, 3.29121971526582136768533121244, 3.35923656130842035513406437440, 3.43860406587292545635236287137, 3.45292607626597335565573062705
Plot not available for L-functions of degree greater than 10.
