| L(s) = 1 | + 2.56e14·2-s − 2.46e30·4-s − 1.16e35·5-s + 2.79e42·7-s − 1.28e45·8-s − 2.98e49·10-s + 9.95e51·11-s − 1.44e56·13-s + 7.18e56·14-s + 5.93e60·16-s − 1.59e62·17-s + 1.62e64·19-s + 2.86e65·20-s + 2.55e66·22-s − 7.65e68·23-s − 2.59e70·25-s − 3.70e70·26-s − 6.90e72·28-s + 7.16e72·29-s + 1.90e75·31-s + 4.78e75·32-s − 4.10e76·34-s − 3.24e77·35-s − 1.10e79·37-s + 4.16e78·38-s + 1.49e80·40-s + 1.87e81·41-s + ⋯ |
| L(s) = 1 | + 0.161·2-s − 0.973·4-s − 0.584·5-s + 0.587·7-s − 0.318·8-s − 0.0943·10-s + 0.255·11-s − 0.804·13-s + 0.0948·14-s + 0.922·16-s − 1.16·17-s + 0.429·19-s + 0.569·20-s + 0.0412·22-s − 1.30·23-s − 0.657·25-s − 0.129·26-s − 0.572·28-s + 0.100·29-s + 0.925·31-s + 0.467·32-s − 0.187·34-s − 0.343·35-s − 0.708·37-s + 0.0692·38-s + 0.186·40-s + 0.673·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(102-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+50.5) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(51)\) |
\(\approx\) |
\(0.4950724695\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4950724695\) |
| \(L(\frac{103}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 - 2.56e14T + 2.53e30T^{2} \) |
| 5 | \( 1 + 1.16e35T + 3.94e70T^{2} \) |
| 7 | \( 1 - 2.79e42T + 2.26e85T^{2} \) |
| 11 | \( 1 - 9.95e51T + 1.51e105T^{2} \) |
| 13 | \( 1 + 1.44e56T + 3.22e112T^{2} \) |
| 17 | \( 1 + 1.59e62T + 1.88e124T^{2} \) |
| 19 | \( 1 - 1.62e64T + 1.42e129T^{2} \) |
| 23 | \( 1 + 7.65e68T + 3.42e137T^{2} \) |
| 29 | \( 1 - 7.16e72T + 5.03e147T^{2} \) |
| 31 | \( 1 - 1.90e75T + 4.24e150T^{2} \) |
| 37 | \( 1 + 1.10e79T + 2.44e158T^{2} \) |
| 41 | \( 1 - 1.87e81T + 7.78e162T^{2} \) |
| 43 | \( 1 - 9.41e81T + 9.55e164T^{2} \) |
| 47 | \( 1 + 3.14e84T + 7.61e168T^{2} \) |
| 53 | \( 1 - 4.00e86T + 1.41e174T^{2} \) |
| 59 | \( 1 - 2.34e89T + 7.17e178T^{2} \) |
| 61 | \( 1 + 4.02e89T + 2.08e180T^{2} \) |
| 67 | \( 1 + 3.05e92T + 2.71e184T^{2} \) |
| 71 | \( 1 - 5.37e93T + 9.48e186T^{2} \) |
| 73 | \( 1 + 2.11e94T + 1.56e188T^{2} \) |
| 79 | \( 1 - 8.91e95T + 4.57e191T^{2} \) |
| 83 | \( 1 + 7.50e96T + 6.71e193T^{2} \) |
| 89 | \( 1 - 3.83e98T + 7.73e196T^{2} \) |
| 97 | \( 1 - 1.03e100T + 4.61e200T^{2} \) |
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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
−9.307328488049180237130780785258, −8.317534233024969817785696401637, −7.62315719476112431868877741789, −6.33131961909452459538532691855, −5.16050567853098106601532275995, −4.41921328545098636902450234530, −3.75018242380532485533038272819, −2.51278267053545977004690260717, −1.41071784755120448793421570898, −0.24436467801514055921314966311,
0.24436467801514055921314966311, 1.41071784755120448793421570898, 2.51278267053545977004690260717, 3.75018242380532485533038272819, 4.41921328545098636902450234530, 5.16050567853098106601532275995, 6.33131961909452459538532691855, 7.62315719476112431868877741789, 8.317534233024969817785696401637, 9.307328488049180237130780785258
