| L(s) = 1 | + 2-s + 1.33·3-s + 4-s + 0.688·5-s + 1.33·6-s − 0.213·7-s + 8-s − 1.20·9-s + 0.688·10-s − 11-s + 1.33·12-s − 3.34·13-s − 0.213·14-s + 0.922·15-s + 16-s + 0.666·17-s − 1.20·18-s + 0.688·20-s − 0.286·21-s − 22-s − 2.71·23-s + 1.33·24-s − 4.52·25-s − 3.34·26-s − 5.63·27-s − 0.213·28-s − 0.247·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.773·3-s + 0.5·4-s + 0.308·5-s + 0.546·6-s − 0.0808·7-s + 0.353·8-s − 0.401·9-s + 0.217·10-s − 0.301·11-s + 0.386·12-s − 0.928·13-s − 0.0571·14-s + 0.238·15-s + 0.250·16-s + 0.161·17-s − 0.284·18-s + 0.154·20-s − 0.0625·21-s − 0.213·22-s − 0.566·23-s + 0.273·24-s − 0.905·25-s − 0.656·26-s − 1.08·27-s − 0.0404·28-s − 0.0459·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 1.33T + 3T^{2} \) |
| 5 | \( 1 - 0.688T + 5T^{2} \) |
| 7 | \( 1 + 0.213T + 7T^{2} \) |
| 13 | \( 1 + 3.34T + 13T^{2} \) |
| 17 | \( 1 - 0.666T + 17T^{2} \) |
| 23 | \( 1 + 2.71T + 23T^{2} \) |
| 29 | \( 1 + 0.247T + 29T^{2} \) |
| 31 | \( 1 + 0.310T + 31T^{2} \) |
| 37 | \( 1 + 4.31T + 37T^{2} \) |
| 41 | \( 1 + 1.50T + 41T^{2} \) |
| 43 | \( 1 + 8.26T + 43T^{2} \) |
| 47 | \( 1 - 10.9T + 47T^{2} \) |
| 53 | \( 1 + 7.44T + 53T^{2} \) |
| 59 | \( 1 - 8.09T + 59T^{2} \) |
| 61 | \( 1 + 5.23T + 61T^{2} \) |
| 67 | \( 1 + 15.3T + 67T^{2} \) |
| 71 | \( 1 - 8.95T + 71T^{2} \) |
| 73 | \( 1 + 5.83T + 73T^{2} \) |
| 79 | \( 1 - 1.22T + 79T^{2} \) |
| 83 | \( 1 - 8.00T + 83T^{2} \) |
| 89 | \( 1 - 1.94T + 89T^{2} \) |
| 97 | \( 1 - 10.7T + 97T^{2} \) |
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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
−7.59679216232539959297046825953, −6.77029322987239070555984614491, −5.98491691114232679951853610660, −5.38991057514676192527924852098, −4.67876709889282401630231789418, −3.77700831186388585892163358247, −3.11938156282125479251501302626, −2.38397544501331625521909519939, −1.72896194501863279465446912566, 0,
1.72896194501863279465446912566, 2.38397544501331625521909519939, 3.11938156282125479251501302626, 3.77700831186388585892163358247, 4.67876709889282401630231789418, 5.38991057514676192527924852098, 5.98491691114232679951853610660, 6.77029322987239070555984614491, 7.59679216232539959297046825953
