| L(s) = 1 | + 1.25e17·2-s + 4.55e27·3-s − 2.58e34·4-s + 2.05e40·5-s + 5.71e44·6-s − 2.84e48·7-s − 8.44e51·8-s + 1.33e55·9-s + 2.57e57·10-s − 9.76e59·11-s − 1.17e62·12-s + 1.01e64·13-s − 3.56e65·14-s + 9.35e67·15-s + 1.27e67·16-s + 2.05e70·17-s + 1.67e72·18-s + 3.04e73·19-s − 5.29e74·20-s − 1.29e76·21-s − 1.22e77·22-s + 2.95e78·23-s − 3.85e79·24-s + 1.80e80·25-s + 1.26e81·26-s + 2.73e82·27-s + 7.33e82·28-s + ⋯ |
| L(s) = 1 | + 0.615·2-s + 1.67·3-s − 0.621·4-s + 1.32·5-s + 1.03·6-s − 0.725·7-s − 0.997·8-s + 1.81·9-s + 0.813·10-s − 1.28·11-s − 1.04·12-s + 0.898·13-s − 0.446·14-s + 2.21·15-s + 0.00737·16-s + 0.364·17-s + 1.11·18-s + 0.902·19-s − 0.821·20-s − 1.21·21-s − 0.792·22-s + 1.48·23-s − 1.67·24-s + 0.749·25-s + 0.553·26-s + 1.35·27-s + 0.450·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(116-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+57.5) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(58)\) |
\(\approx\) |
\(5.813412329\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.813412329\) |
| \(L(\frac{117}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 - 1.25e17T + 4.15e34T^{2} \) |
| 3 | \( 1 - 4.55e27T + 7.39e54T^{2} \) |
| 5 | \( 1 - 2.05e40T + 2.40e80T^{2} \) |
| 7 | \( 1 + 2.84e48T + 1.53e97T^{2} \) |
| 11 | \( 1 + 9.76e59T + 5.75e119T^{2} \) |
| 13 | \( 1 - 1.01e64T + 1.26e128T^{2} \) |
| 17 | \( 1 - 2.05e70T + 3.17e141T^{2} \) |
| 19 | \( 1 - 3.04e73T + 1.13e147T^{2} \) |
| 23 | \( 1 - 2.95e78T + 3.96e156T^{2} \) |
| 29 | \( 1 - 1.02e84T + 1.49e168T^{2} \) |
| 31 | \( 1 - 1.10e86T + 3.21e171T^{2} \) |
| 37 | \( 1 - 1.90e90T + 2.20e180T^{2} \) |
| 41 | \( 1 - 5.55e92T + 2.95e185T^{2} \) |
| 43 | \( 1 + 1.95e93T + 7.06e187T^{2} \) |
| 47 | \( 1 + 1.92e96T + 1.95e192T^{2} \) |
| 53 | \( 1 + 1.77e99T + 1.95e198T^{2} \) |
| 59 | \( 1 - 1.68e101T + 4.44e203T^{2} \) |
| 61 | \( 1 + 2.99e102T + 2.05e205T^{2} \) |
| 67 | \( 1 + 3.49e104T + 9.96e209T^{2} \) |
| 71 | \( 1 - 5.93e105T + 7.84e212T^{2} \) |
| 73 | \( 1 + 7.68e106T + 1.91e214T^{2} \) |
| 79 | \( 1 + 1.57e109T + 1.68e218T^{2} \) |
| 83 | \( 1 - 2.04e110T + 4.94e220T^{2} \) |
| 89 | \( 1 - 1.39e112T + 1.51e224T^{2} \) |
| 97 | \( 1 - 6.91e111T + 3.01e228T^{2} \) |
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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
−13.38761074154440607828278396132, −13.06990208071493889256481727535, −10.00327508623106896880504381476, −9.268481193527474835328803845475, −8.085428695595610252089535445768, −6.19087352258432492176616968787, −4.81198410869402393096820717949, −3.15410569298235779071180274854, −2.73756399525181738047914681683, −1.09688694408566665696505383861,
1.09688694408566665696505383861, 2.73756399525181738047914681683, 3.15410569298235779071180274854, 4.81198410869402393096820717949, 6.19087352258432492176616968787, 8.085428695595610252089535445768, 9.268481193527474835328803845475, 10.00327508623106896880504381476, 13.06990208071493889256481727535, 13.38761074154440607828278396132
