| L(s) = 1 | + 1.02·2-s − 8.89·3-s − 30.9·4-s + 25·5-s − 9.09·6-s − 228.·7-s − 64.4·8-s − 163.·9-s + 25.5·10-s + 738.·11-s + 275.·12-s + 207.·13-s − 233.·14-s − 222.·15-s + 924.·16-s − 784.·17-s − 167.·18-s − 627.·19-s − 773.·20-s + 2.03e3·21-s + 755.·22-s − 529·23-s + 572.·24-s + 625·25-s + 212.·26-s + 3.61e3·27-s + 7.06e3·28-s + ⋯ |
| L(s) = 1 | + 0.180·2-s − 0.570·3-s − 0.967·4-s + 0.447·5-s − 0.103·6-s − 1.76·7-s − 0.355·8-s − 0.674·9-s + 0.0808·10-s + 1.83·11-s + 0.551·12-s + 0.341·13-s − 0.318·14-s − 0.255·15-s + 0.902·16-s − 0.658·17-s − 0.121·18-s − 0.398·19-s − 0.432·20-s + 1.00·21-s + 0.332·22-s − 0.208·23-s + 0.202·24-s + 0.200·25-s + 0.0617·26-s + 0.955·27-s + 1.70·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.9588054262\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9588054262\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 25T \) |
| 23 | \( 1 + 529T \) |
| good | 2 | \( 1 - 1.02T + 32T^{2} \) |
| 3 | \( 1 + 8.89T + 243T^{2} \) |
| 7 | \( 1 + 228.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 738.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 207.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 784.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 627.T + 2.47e6T^{2} \) |
| 29 | \( 1 - 7.50e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 4.90e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.49e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 6.89e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.13e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.58e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 8.82e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.91e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.40e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.96e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.21e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.66e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.79e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.19e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.18e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.47e5T + 8.58e9T^{2} \) |
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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
−12.62851554324833761553902281987, −11.85542724243928944662158068471, −10.36675656468956988855316585179, −9.353106554025425978194842241522, −8.734568815384490264077941788009, −6.43972724365386908213679405641, −6.13950009643455804735101247030, −4.43444132854155837712292125853, −3.17658714567202750440037304184, −0.68219626799653917970463596921,
0.68219626799653917970463596921, 3.17658714567202750440037304184, 4.43444132854155837712292125853, 6.13950009643455804735101247030, 6.43972724365386908213679405641, 8.734568815384490264077941788009, 9.353106554025425978194842241522, 10.36675656468956988855316585179, 11.85542724243928944662158068471, 12.62851554324833761553902281987
