| L(s) = 1 | − 3·2-s + 3·4-s + 3·5-s − 2·7-s + 2·8-s + 2·9-s − 9·10-s + 10·11-s − 3·13-s + 6·14-s − 9·16-s + 2·17-s − 6·18-s + 9·19-s + 9·20-s − 30·22-s − 22·23-s + 3·25-s + 9·26-s + 4·27-s − 6·28-s − 4·29-s − 24·31-s + 9·32-s − 6·34-s − 6·35-s + 6·36-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 3/2·4-s + 1.34·5-s − 0.755·7-s + 0.707·8-s + 2/3·9-s − 2.84·10-s + 3.01·11-s − 0.832·13-s + 1.60·14-s − 9/4·16-s + 0.485·17-s − 1.41·18-s + 2.06·19-s + 2.01·20-s − 6.39·22-s − 4.58·23-s + 3/5·25-s + 1.76·26-s + 0.769·27-s − 1.13·28-s − 0.742·29-s − 4.31·31-s + 1.59·32-s − 1.02·34-s − 1.01·35-s + 36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{6} \cdot 37^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{6} \cdot 37^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2858559877\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2858559877\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T + T^{2} )^{3} \) |
| 5 | \( ( 1 - T + T^{2} )^{3} \) |
| 37 | \( 1 - 9 T + 72 T^{2} - 403 T^{3} + 72 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| good | 3 | \( 1 - 2 T^{2} - 4 T^{3} - 2 T^{4} + 4 T^{5} + 46 T^{6} + 4 p T^{7} - 2 p^{2} T^{8} - 4 p^{3} T^{9} - 2 p^{4} T^{10} + p^{6} T^{12} \) |
| 7 | \( 1 + 2 T - 3 T^{2} + 22 T^{3} + 8 T^{4} - 18 p T^{5} + 163 T^{6} - 18 p^{2} T^{7} + 8 p^{2} T^{8} + 22 p^{3} T^{9} - 3 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \) |
| 11 | \( ( 1 - 5 T + 37 T^{2} - 108 T^{3} + 37 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 13 | \( 1 + 3 T + T^{2} + 86 T^{3} + 53 T^{4} - 365 T^{5} + 3002 T^{6} - 365 p T^{7} + 53 p^{2} T^{8} + 86 p^{3} T^{9} + p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 17 | \( 1 - 2 T + 7 T^{2} - 210 T^{3} + 276 T^{4} - 1106 T^{5} + 20181 T^{6} - 1106 p T^{7} + 276 p^{2} T^{8} - 210 p^{3} T^{9} + 7 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \) |
| 19 | \( 1 - 9 T + 4 T^{2} - T^{3} + 1352 T^{4} - 3557 T^{5} - 8458 T^{6} - 3557 p T^{7} + 1352 p^{2} T^{8} - p^{3} T^{9} + 4 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \) |
| 23 | \( ( 1 + 11 T + 105 T^{2} + 540 T^{3} + 105 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 29 | \( ( 1 + 2 T + 59 T^{2} + 124 T^{3} + 59 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( ( 1 + 12 T + 102 T^{2} + 636 T^{3} + 102 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 41 | \( 1 + 12 T - 20 T^{2} - 68 T^{3} + 7956 T^{4} + 628 p T^{5} - 170634 T^{6} + 628 p^{2} T^{7} + 7956 p^{2} T^{8} - 68 p^{3} T^{9} - 20 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \) |
| 43 | \( ( 1 + 8 T + 118 T^{2} + 686 T^{3} + 118 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 47 | \( ( 1 - 17 T + 3 p T^{2} - 876 T^{3} + 3 p^{2} T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 53 | \( 1 + 6 T - 92 T^{2} - 436 T^{3} + 5988 T^{4} + 12502 T^{5} - 309318 T^{6} + 12502 p T^{7} + 5988 p^{2} T^{8} - 436 p^{3} T^{9} - 92 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 59 | \( 1 - 13 T - 24 T^{2} + 491 T^{3} + 6362 T^{4} - 38691 T^{5} - 125222 T^{6} - 38691 p T^{7} + 6362 p^{2} T^{8} + 491 p^{3} T^{9} - 24 p^{4} T^{10} - 13 p^{5} T^{11} + p^{6} T^{12} \) |
| 61 | \( 1 - 26 T + 297 T^{2} - 2662 T^{3} + 27026 T^{4} - 257442 T^{5} + 2107681 T^{6} - 257442 p T^{7} + 27026 p^{2} T^{8} - 2662 p^{3} T^{9} + 297 p^{4} T^{10} - 26 p^{5} T^{11} + p^{6} T^{12} \) |
| 67 | \( 1 + 8 T - 29 T^{2} + 296 T^{3} + 626 T^{4} - 42952 T^{5} - 215077 T^{6} - 42952 p T^{7} + 626 p^{2} T^{8} + 296 p^{3} T^{9} - 29 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \) |
| 71 | \( 1 + 10 T + 15 T^{2} - 38 T^{3} - 3250 T^{4} - 32550 T^{5} - 104117 T^{6} - 32550 p T^{7} - 3250 p^{2} T^{8} - 38 p^{3} T^{9} + 15 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} \) |
| 73 | \( ( 1 - 2 T + 165 T^{2} - 116 T^{3} + 165 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 79 | \( 1 + 16 T - 49 T^{2} - 304 T^{3} + 27458 T^{4} + 119376 T^{5} - 1242129 T^{6} + 119376 p T^{7} + 27458 p^{2} T^{8} - 304 p^{3} T^{9} - 49 p^{4} T^{10} + 16 p^{5} T^{11} + p^{6} T^{12} \) |
| 83 | \( 1 - 10 T - 117 T^{2} + 446 T^{3} + 15998 T^{4} + 7494 T^{5} - 1742849 T^{6} + 7494 p T^{7} + 15998 p^{2} T^{8} + 446 p^{3} T^{9} - 117 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} \) |
| 89 | \( 1 - 27 T + 282 T^{2} - 2025 T^{3} + 24432 T^{4} - 310419 T^{5} + 3073012 T^{6} - 310419 p T^{7} + 24432 p^{2} T^{8} - 2025 p^{3} T^{9} + 282 p^{4} T^{10} - 27 p^{5} T^{11} + p^{6} T^{12} \) |
| 97 | \( ( 1 - 6 T + 167 T^{2} - 292 T^{3} + 167 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.27278864180423343487412863745, −6.23509721058419002339828757510, −5.75348154401639637230193194635, −5.68066756347350282256813898930, −5.46510240700244332980823877952, −5.45000991581499805467339503962, −5.31554251265456536822660981179, −4.91012851272166238923473581858, −4.86296514984672649015973965207, −4.37583017006602314302165414779, −4.07845937619389538444402386742, −3.91203350871080451544902344016, −3.89962660111783910739408884010, −3.67370195788944563763667547806, −3.62179136425875524702734877258, −3.57125942676901183339414649271, −2.78284069735148578753788491084, −2.41405537127740489475931981956, −2.40628554290662796725594291143, −2.08998491759870188754586170695, −1.84914645517564051397878052737, −1.35353940923775450100826079347, −1.30779729684796337297059234854, −1.19440734420602831981153952535, −0.22773485841677310350487891455,
0.22773485841677310350487891455, 1.19440734420602831981153952535, 1.30779729684796337297059234854, 1.35353940923775450100826079347, 1.84914645517564051397878052737, 2.08998491759870188754586170695, 2.40628554290662796725594291143, 2.41405537127740489475931981956, 2.78284069735148578753788491084, 3.57125942676901183339414649271, 3.62179136425875524702734877258, 3.67370195788944563763667547806, 3.89962660111783910739408884010, 3.91203350871080451544902344016, 4.07845937619389538444402386742, 4.37583017006602314302165414779, 4.86296514984672649015973965207, 4.91012851272166238923473581858, 5.31554251265456536822660981179, 5.45000991581499805467339503962, 5.46510240700244332980823877952, 5.68066756347350282256813898930, 5.75348154401639637230193194635, 6.23509721058419002339828757510, 6.27278864180423343487412863745
Plot not available for L-functions of degree greater than 10.
