| L(s) = 1 | + 6·3-s + 5-s − 5·7-s + 21·9-s − 6·11-s + 6·15-s + 9·17-s + 7·19-s − 30·21-s + 12·23-s − 10·25-s + 56·27-s + 7·29-s − 11·31-s − 36·33-s − 5·35-s + 6·37-s + 13·41-s + 15·43-s + 21·45-s − 9·47-s − 2·49-s + 54·51-s + 22·53-s − 6·55-s + 42·57-s − 7·59-s + ⋯ |
| L(s) = 1 | + 3.46·3-s + 0.447·5-s − 1.88·7-s + 7·9-s − 1.80·11-s + 1.54·15-s + 2.18·17-s + 1.60·19-s − 6.54·21-s + 2.50·23-s − 2·25-s + 10.7·27-s + 1.29·29-s − 1.97·31-s − 6.26·33-s − 0.845·35-s + 0.986·37-s + 2.03·41-s + 2.28·43-s + 3.13·45-s − 1.31·47-s − 2/7·49-s + 7.56·51-s + 3.02·53-s − 0.809·55-s + 5.56·57-s − 0.911·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{6} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{6} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(89.54392483\) |
| \(L(\frac12)\) |
\(\approx\) |
\(89.54392483\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 - T )^{6} \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - T + 11 T^{2} - 8 T^{3} + 98 T^{4} - 66 T^{5} + 553 T^{6} - 66 p T^{7} + 98 p^{2} T^{8} - 8 p^{3} T^{9} + 11 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \) |
| 7 | \( 1 + 5 T + 27 T^{2} + 106 T^{3} + 54 p T^{4} + 160 p T^{5} + 463 p T^{6} + 160 p^{2} T^{7} + 54 p^{3} T^{8} + 106 p^{3} T^{9} + 27 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \) |
| 11 | \( 1 + 6 T + 43 T^{2} + 212 T^{3} + 929 T^{4} + 3478 T^{5} + 12603 T^{6} + 3478 p T^{7} + 929 p^{2} T^{8} + 212 p^{3} T^{9} + 43 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 17 | \( 1 - 9 T + 114 T^{2} - 716 T^{3} + 5091 T^{4} - 23603 T^{5} + 117036 T^{6} - 23603 p T^{7} + 5091 p^{2} T^{8} - 716 p^{3} T^{9} + 114 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \) |
| 19 | \( 1 - 7 T + 94 T^{2} - 504 T^{3} + 3907 T^{4} - 16989 T^{5} + 95148 T^{6} - 16989 p T^{7} + 3907 p^{2} T^{8} - 504 p^{3} T^{9} + 94 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \) |
| 23 | \( 1 - 12 T + 123 T^{2} - 807 T^{3} + 5631 T^{4} - 30803 T^{5} + 170082 T^{6} - 30803 p T^{7} + 5631 p^{2} T^{8} - 807 p^{3} T^{9} + 123 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \) |
| 29 | \( 1 - 7 T + 3 p T^{2} - 308 T^{3} + 3534 T^{4} - 13160 T^{5} + 134813 T^{6} - 13160 p T^{7} + 3534 p^{2} T^{8} - 308 p^{3} T^{9} + 3 p^{5} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \) |
| 31 | \( 1 + 11 T + 203 T^{2} + 50 p T^{3} + 15928 T^{4} + 90560 T^{5} + 656651 T^{6} + 90560 p T^{7} + 15928 p^{2} T^{8} + 50 p^{4} T^{9} + 203 p^{4} T^{10} + 11 p^{5} T^{11} + p^{6} T^{12} \) |
| 37 | \( 1 - 6 T + 133 T^{2} - 427 T^{3} + 6955 T^{4} - 9763 T^{5} + 252486 T^{6} - 9763 p T^{7} + 6955 p^{2} T^{8} - 427 p^{3} T^{9} + 133 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 41 | \( 1 - 13 T + 130 T^{2} - 992 T^{3} + 8541 T^{4} - 49751 T^{5} + 322416 T^{6} - 49751 p T^{7} + 8541 p^{2} T^{8} - 992 p^{3} T^{9} + 130 p^{4} T^{10} - 13 p^{5} T^{11} + p^{6} T^{12} \) |
| 43 | \( 1 - 15 T + 141 T^{2} - 1430 T^{3} + 14903 T^{4} - 106255 T^{5} + 669262 T^{6} - 106255 p T^{7} + 14903 p^{2} T^{8} - 1430 p^{3} T^{9} + 141 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \) |
| 47 | \( 1 + 9 T + 2 p T^{2} - 114 T^{3} + 643 T^{4} - 14423 T^{5} + 140348 T^{6} - 14423 p T^{7} + 643 p^{2} T^{8} - 114 p^{3} T^{9} + 2 p^{5} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \) |
| 53 | \( ( 1 - 11 T + 183 T^{2} - 1179 T^{3} + 183 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 59 | \( 1 + 7 T + 204 T^{2} + 238 T^{3} + 10077 T^{4} - 87493 T^{5} + 180708 T^{6} - 87493 p T^{7} + 10077 p^{2} T^{8} + 238 p^{3} T^{9} + 204 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \) |
| 61 | \( 1 - 25 T + 440 T^{2} - 5900 T^{3} + 66737 T^{4} - 638391 T^{5} + 5343900 T^{6} - 638391 p T^{7} + 66737 p^{2} T^{8} - 5900 p^{3} T^{9} + 440 p^{4} T^{10} - 25 p^{5} T^{11} + p^{6} T^{12} \) |
| 67 | \( 1 - 5 T + 226 T^{2} + 354 T^{3} + 12789 T^{4} + 190155 T^{5} + 293336 T^{6} + 190155 p T^{7} + 12789 p^{2} T^{8} + 354 p^{3} T^{9} + 226 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \) |
| 71 | \( 1 - 8 T + 281 T^{2} - 2669 T^{3} + 38549 T^{4} - 365241 T^{5} + 3333546 T^{6} - 365241 p T^{7} + 38549 p^{2} T^{8} - 2669 p^{3} T^{9} + 281 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \) |
| 73 | \( 1 - 15 T + 3 p T^{2} - 1456 T^{3} + 9232 T^{4} + 19750 T^{5} - 99129 T^{6} + 19750 p T^{7} + 9232 p^{2} T^{8} - 1456 p^{3} T^{9} + 3 p^{5} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \) |
| 79 | \( 1 - 14 T + 145 T^{2} - 616 T^{3} + 11799 T^{4} - 137088 T^{5} + 1824577 T^{6} - 137088 p T^{7} + 11799 p^{2} T^{8} - 616 p^{3} T^{9} + 145 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} \) |
| 83 | \( 1 + 13 T + 221 T^{2} + 1524 T^{3} + 20630 T^{4} + 159630 T^{5} + 2122165 T^{6} + 159630 p T^{7} + 20630 p^{2} T^{8} + 1524 p^{3} T^{9} + 221 p^{4} T^{10} + 13 p^{5} T^{11} + p^{6} T^{12} \) |
| 89 | \( 1 - 33 T + 710 T^{2} - 10824 T^{3} + 141665 T^{4} - 1595947 T^{5} + 16164832 T^{6} - 1595947 p T^{7} + 141665 p^{2} T^{8} - 10824 p^{3} T^{9} + 710 p^{4} T^{10} - 33 p^{5} T^{11} + p^{6} T^{12} \) |
| 97 | \( 1 - 50 T + 1348 T^{2} - 25615 T^{3} + 388179 T^{4} - 4903428 T^{5} + 52476469 T^{6} - 4903428 p T^{7} + 388179 p^{2} T^{8} - 25615 p^{3} T^{9} + 1348 p^{4} T^{10} - 50 p^{5} T^{11} + p^{6} T^{12} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.42556917918469450587392178877, −3.86125699139085715847551905669, −3.80311823292674630030361916110, −3.75031715504071794551370802059, −3.72743283637119550623362156495, −3.56170915418439695917379750530, −3.50658588176978326064733777632, −3.19831522720164251665746945645, −3.16529077725331483658634918040, −3.03472679090766947691395538016, −2.84882179731929268017698448928, −2.82368236504562179420383336656, −2.74401980514231676635807727137, −2.37081275213468629777437291038, −2.21955025030583099581149054599, −2.18685624984650365273124178163, −1.99192125140785734051341695848, −1.93992560467405798011821839612, −1.86312191754174594161417444205, −1.26540526978308159887043557136, −1.13418000168854840165062605301, −0.848993603573427775272206251365, −0.789997458263512926483671216196, −0.63135110701700118266770776544, −0.53358432102264110531864649662,
0.53358432102264110531864649662, 0.63135110701700118266770776544, 0.789997458263512926483671216196, 0.848993603573427775272206251365, 1.13418000168854840165062605301, 1.26540526978308159887043557136, 1.86312191754174594161417444205, 1.93992560467405798011821839612, 1.99192125140785734051341695848, 2.18685624984650365273124178163, 2.21955025030583099581149054599, 2.37081275213468629777437291038, 2.74401980514231676635807727137, 2.82368236504562179420383336656, 2.84882179731929268017698448928, 3.03472679090766947691395538016, 3.16529077725331483658634918040, 3.19831522720164251665746945645, 3.50658588176978326064733777632, 3.56170915418439695917379750530, 3.72743283637119550623362156495, 3.75031715504071794551370802059, 3.80311823292674630030361916110, 3.86125699139085715847551905669, 4.42556917918469450587392178877
Plot not available for L-functions of degree greater than 10.
