| L(s) = 1 | − 2.45·2-s − 9·3-s − 25.9·4-s − 25·5-s + 22.1·6-s + 16.3·7-s + 142.·8-s + 81·9-s + 61.4·10-s − 85.4·11-s + 233.·12-s − 169·13-s − 40.2·14-s + 225·15-s + 480.·16-s + 1.00e3·17-s − 199.·18-s + 2.41e3·19-s + 649.·20-s − 147.·21-s + 209.·22-s − 18.4·23-s − 1.28e3·24-s + 625·25-s + 415.·26-s − 729·27-s − 425.·28-s + ⋯ |
| L(s) = 1 | − 0.434·2-s − 0.577·3-s − 0.811·4-s − 0.447·5-s + 0.250·6-s + 0.126·7-s + 0.786·8-s + 0.333·9-s + 0.194·10-s − 0.212·11-s + 0.468·12-s − 0.277·13-s − 0.0548·14-s + 0.258·15-s + 0.469·16-s + 0.844·17-s − 0.144·18-s + 1.53·19-s + 0.362·20-s − 0.0729·21-s + 0.0924·22-s − 0.00727·23-s − 0.454·24-s + 0.200·25-s + 0.120·26-s − 0.192·27-s − 0.102·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 9T \) |
| 5 | \( 1 + 25T \) |
| 13 | \( 1 + 169T \) |
| good | 2 | \( 1 + 2.45T + 32T^{2} \) |
| 7 | \( 1 - 16.3T + 1.68e4T^{2} \) |
| 11 | \( 1 + 85.4T + 1.61e5T^{2} \) |
| 17 | \( 1 - 1.00e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.41e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 18.4T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.50e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.81e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 3.75e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 9.44e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 3.43e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.18e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 8.39e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 65.9T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.97e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.10e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.03e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.96e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 5.27e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.26e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 9.09e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.57e5T + 8.58e9T^{2} \) |
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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
−11.08506852751874856548647136265, −10.03810988068558443855200685199, −9.272011898561558702603474267889, −8.004947215077619461313369791850, −7.29347538015409845030474883584, −5.62976706324293134812495870633, −4.75201778408806098263205738561, −3.44061472984834796424891646087, −1.24620973085343393094717005698, 0,
1.24620973085343393094717005698, 3.44061472984834796424891646087, 4.75201778408806098263205738561, 5.62976706324293134812495870633, 7.29347538015409845030474883584, 8.004947215077619461313369791850, 9.272011898561558702603474267889, 10.03810988068558443855200685199, 11.08506852751874856548647136265
