| L(s) = 1 | + 2-s + 0.347·3-s + 4-s + 0.347·6-s − 7-s + 8-s − 2.87·9-s + 4.29·11-s + 0.347·12-s + 0.532·13-s − 14-s + 16-s + 17-s − 2.87·18-s − 4.41·19-s − 0.347·21-s + 4.29·22-s − 7.86·23-s + 0.347·24-s + 0.532·26-s − 2.04·27-s − 28-s − 3.57·29-s − 9.17·31-s + 32-s + 1.49·33-s + 34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.200·3-s + 0.5·4-s + 0.141·6-s − 0.377·7-s + 0.353·8-s − 0.959·9-s + 1.29·11-s + 0.100·12-s + 0.147·13-s − 0.267·14-s + 0.250·16-s + 0.242·17-s − 0.678·18-s − 1.01·19-s − 0.0757·21-s + 0.914·22-s − 1.63·23-s + 0.0708·24-s + 0.104·26-s − 0.392·27-s − 0.188·28-s − 0.663·29-s − 1.64·31-s + 0.176·32-s + 0.259·33-s + 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 - 0.347T + 3T^{2} \) |
| 11 | \( 1 - 4.29T + 11T^{2} \) |
| 13 | \( 1 - 0.532T + 13T^{2} \) |
| 19 | \( 1 + 4.41T + 19T^{2} \) |
| 23 | \( 1 + 7.86T + 23T^{2} \) |
| 29 | \( 1 + 3.57T + 29T^{2} \) |
| 31 | \( 1 + 9.17T + 31T^{2} \) |
| 37 | \( 1 - 0.389T + 37T^{2} \) |
| 41 | \( 1 - 1.65T + 41T^{2} \) |
| 43 | \( 1 + 4.83T + 43T^{2} \) |
| 47 | \( 1 - 8.78T + 47T^{2} \) |
| 53 | \( 1 + 8.49T + 53T^{2} \) |
| 59 | \( 1 + 14.3T + 59T^{2} \) |
| 61 | \( 1 + 1.77T + 61T^{2} \) |
| 67 | \( 1 + 3.95T + 67T^{2} \) |
| 71 | \( 1 - 3.55T + 71T^{2} \) |
| 73 | \( 1 - 11.3T + 73T^{2} \) |
| 79 | \( 1 - 2.04T + 79T^{2} \) |
| 83 | \( 1 - 4.02T + 83T^{2} \) |
| 89 | \( 1 - 6.29T + 89T^{2} \) |
| 97 | \( 1 + 8.04T + 97T^{2} \) |
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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
−7.71022275852719075994285672344, −6.85221716075215759764649308422, −6.07270736046560279197831685885, −5.84656213605026144026722875184, −4.75705838964889445398017166017, −3.82196088301592266338963242936, −3.53147318800763582461152425030, −2.39969272249729382367884235161, −1.62780525439080638227539515838, 0,
1.62780525439080638227539515838, 2.39969272249729382367884235161, 3.53147318800763582461152425030, 3.82196088301592266338963242936, 4.75705838964889445398017166017, 5.84656213605026144026722875184, 6.07270736046560279197831685885, 6.85221716075215759764649308422, 7.71022275852719075994285672344
