| L(s) = 1 | − 468·3-s + 5.62e4·5-s − 3.33e5·7-s − 1.37e6·9-s + 6.39e6·11-s + 1.51e7·13-s − 2.63e7·15-s + 4.31e7·17-s + 3.65e8·19-s + 1.55e8·21-s + 5.72e7·23-s + 1.93e9·25-s + 1.38e9·27-s − 4.64e7·29-s + 5.68e9·31-s − 2.99e9·33-s − 1.87e10·35-s − 1.88e9·37-s − 7.11e9·39-s − 7.33e9·41-s + 2.68e10·43-s − 7.73e10·45-s − 1.01e11·47-s + 1.40e10·49-s − 2.01e10·51-s + 2.78e11·53-s + 3.59e11·55-s + ⋯ |
| L(s) = 1 | − 0.370·3-s + 1.60·5-s − 1.06·7-s − 0.862·9-s + 1.08·11-s + 0.873·13-s − 0.596·15-s + 0.433·17-s + 1.78·19-s + 0.396·21-s + 0.0806·23-s + 1.58·25-s + 0.690·27-s − 0.0144·29-s + 1.14·31-s − 0.403·33-s − 1.72·35-s − 0.120·37-s − 0.323·39-s − 0.241·41-s + 0.648·43-s − 1.38·45-s − 1.37·47-s + 0.144·49-s − 0.160·51-s + 1.72·53-s + 1.75·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(2.061015010\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.061015010\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + 52 p^{2} T + p^{13} T^{2} \) |
| 5 | \( 1 - 56214 T + p^{13} T^{2} \) |
| 7 | \( 1 + 47576 p T + p^{13} T^{2} \) |
| 11 | \( 1 - 581580 p T + p^{13} T^{2} \) |
| 13 | \( 1 - 15199742 T + p^{13} T^{2} \) |
| 17 | \( 1 - 43114194 T + p^{13} T^{2} \) |
| 19 | \( 1 - 365115484 T + p^{13} T^{2} \) |
| 23 | \( 1 - 57226824 T + p^{13} T^{2} \) |
| 29 | \( 1 + 46418994 T + p^{13} T^{2} \) |
| 31 | \( 1 - 5682185824 T + p^{13} T^{2} \) |
| 37 | \( 1 + 1887185098 T + p^{13} T^{2} \) |
| 41 | \( 1 + 7336802934 T + p^{13} T^{2} \) |
| 43 | \( 1 - 26886674980 T + p^{13} T^{2} \) |
| 47 | \( 1 + 101839834224 T + p^{13} T^{2} \) |
| 53 | \( 1 - 278731884294 T + p^{13} T^{2} \) |
| 59 | \( 1 + 59573945772 T + p^{13} T^{2} \) |
| 61 | \( 1 + 27484470418 T + p^{13} T^{2} \) |
| 67 | \( 1 + 784410054932 T + p^{13} T^{2} \) |
| 71 | \( 1 - 360365227992 T + p^{13} T^{2} \) |
| 73 | \( 1 + 1592635413718 T + p^{13} T^{2} \) |
| 79 | \( 1 - 23161184752 T + p^{13} T^{2} \) |
| 83 | \( 1 + 2050158110436 T + p^{13} T^{2} \) |
| 89 | \( 1 + 3485391237126 T + p^{13} T^{2} \) |
| 97 | \( 1 - 6706667416802 T + p^{13} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.23758416912153835465560419383, −14.23321319311047109636902744506, −13.37881483256668419541963787746, −11.76890280447658187313306991297, −10.05485923595700085121969096645, −9.063165499557691097016506911780, −6.48812579958183135764738052696, −5.62703441429249929736842833659, −3.08265160901573935908673307852, −1.13979612295243050746634422779,
1.13979612295243050746634422779, 3.08265160901573935908673307852, 5.62703441429249929736842833659, 6.48812579958183135764738052696, 9.063165499557691097016506911780, 10.05485923595700085121969096645, 11.76890280447658187313306991297, 13.37881483256668419541963787746, 14.23321319311047109636902744506, 16.23758416912153835465560419383
