| L(s) = 1 | − 5.93e14·2-s − 2.18e30·4-s − 2.07e34·5-s − 6.15e42·7-s + 2.80e45·8-s + 1.23e49·10-s + 1.93e51·11-s + 2.07e56·13-s + 3.65e57·14-s + 3.87e60·16-s − 9.51e61·17-s − 2.07e64·19-s + 4.53e64·20-s − 1.14e66·22-s + 1.01e69·23-s − 3.90e70·25-s − 1.23e71·26-s + 1.34e73·28-s + 1.12e74·29-s − 3.39e75·31-s − 9.39e75·32-s + 5.64e76·34-s + 1.27e77·35-s − 2.80e78·37-s + 1.23e79·38-s − 5.81e79·40-s + 2.18e81·41-s + ⋯ |
| L(s) = 1 | − 0.372·2-s − 0.861·4-s − 0.104·5-s − 1.29·7-s + 0.693·8-s + 0.0389·10-s + 0.0497·11-s + 1.15·13-s + 0.482·14-s + 0.602·16-s − 0.692·17-s − 0.549·19-s + 0.0900·20-s − 0.0185·22-s + 1.72·23-s − 0.989·25-s − 0.431·26-s + 1.11·28-s + 1.57·29-s − 1.64·31-s − 0.918·32-s + 0.258·34-s + 0.135·35-s − 0.179·37-s + 0.204·38-s − 0.0725·40-s + 0.783·41-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
| \(L(51)\) |
\(\approx\) |
\(1.081215259\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.081215259\) |
| \(L(\frac{103}{2})\) |
|
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 | \( 1 + 5.93e14T + 2.53e30T^{2} \) |
| 5 | \( 1 + 2.07e34T + 3.94e70T^{2} \) |
| 7 | \( 1 + 6.15e42T + 2.26e85T^{2} \) |
| 11 | \( 1 - 1.93e51T + 1.51e105T^{2} \) |
| 13 | \( 1 - 2.07e56T + 3.22e112T^{2} \) |
| 17 | \( 1 + 9.51e61T + 1.88e124T^{2} \) |
| 19 | \( 1 + 2.07e64T + 1.42e129T^{2} \) |
| 23 | \( 1 - 1.01e69T + 3.42e137T^{2} \) |
| 29 | \( 1 - 1.12e74T + 5.03e147T^{2} \) |
| 31 | \( 1 + 3.39e75T + 4.24e150T^{2} \) |
| 37 | \( 1 + 2.80e78T + 2.44e158T^{2} \) |
| 41 | \( 1 - 2.18e81T + 7.78e162T^{2} \) |
| 43 | \( 1 - 4.27e82T + 9.55e164T^{2} \) |
| 47 | \( 1 - 4.06e84T + 7.61e168T^{2} \) |
| 53 | \( 1 - 1.28e87T + 1.41e174T^{2} \) |
| 59 | \( 1 + 2.29e88T + 7.17e178T^{2} \) |
| 61 | \( 1 - 1.48e90T + 2.08e180T^{2} \) |
| 67 | \( 1 + 1.12e92T + 2.71e184T^{2} \) |
| 71 | \( 1 - 1.07e93T + 9.48e186T^{2} \) |
| 73 | \( 1 - 8.88e92T + 1.56e188T^{2} \) |
| 79 | \( 1 - 3.94e95T + 4.57e191T^{2} \) |
| 83 | \( 1 + 7.91e96T + 6.71e193T^{2} \) |
| 89 | \( 1 + 1.56e98T + 7.73e196T^{2} \) |
| 97 | \( 1 - 6.75e99T + 4.61e200T^{2} \) |
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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
−9.101045612022318937045049014536, −8.776256639844815329899402038617, −7.42931687922882984773698606323, −6.43550756861475922214417643220, −5.49648370967442618954376144418, −4.23385647855800077260159831180, −3.61121935376204598056546093772, −2.52167039032803704821656359287, −1.14502648709429845302252731170, −0.44899650814103832743841340268,
0.44899650814103832743841340268, 1.14502648709429845302252731170, 2.52167039032803704821656359287, 3.61121935376204598056546093772, 4.23385647855800077260159831180, 5.49648370967442618954376144418, 6.43550756861475922214417643220, 7.42931687922882984773698606323, 8.776256639844815329899402038617, 9.101045612022318937045049014536
