| L(s) = 1 | + 2-s − 0.618·3-s + 4-s + 3.85·5-s − 0.618·6-s − 1.23·7-s + 8-s − 2.61·9-s + 3.85·10-s + 11-s − 0.618·12-s + 0.763·13-s − 1.23·14-s − 2.38·15-s + 16-s − 2.47·17-s − 2.61·18-s + 3.85·20-s + 0.763·21-s + 22-s + 3.85·23-s − 0.618·24-s + 9.85·25-s + 0.763·26-s + 3.47·27-s − 1.23·28-s + 1.52·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.356·3-s + 0.5·4-s + 1.72·5-s − 0.252·6-s − 0.467·7-s + 0.353·8-s − 0.872·9-s + 1.21·10-s + 0.301·11-s − 0.178·12-s + 0.211·13-s − 0.330·14-s − 0.615·15-s + 0.250·16-s − 0.599·17-s − 0.617·18-s + 0.861·20-s + 0.166·21-s + 0.213·22-s + 0.803·23-s − 0.126·24-s + 1.97·25-s + 0.149·26-s + 0.668·27-s − 0.233·28-s + 0.283·29-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
| \(L(1)\) |
\(\approx\) |
\(3.893770328\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.893770328\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.618T + 3T^{2} \) |
| 5 | \( 1 - 3.85T + 5T^{2} \) |
| 7 | \( 1 + 1.23T + 7T^{2} \) |
| 13 | \( 1 - 0.763T + 13T^{2} \) |
| 17 | \( 1 + 2.47T + 17T^{2} \) |
| 23 | \( 1 - 3.85T + 23T^{2} \) |
| 29 | \( 1 - 1.52T + 29T^{2} \) |
| 31 | \( 1 - 5.09T + 31T^{2} \) |
| 37 | \( 1 - 7.61T + 37T^{2} \) |
| 41 | \( 1 - 6.94T + 41T^{2} \) |
| 43 | \( 1 - 1.23T + 43T^{2} \) |
| 47 | \( 1 + 8.09T + 47T^{2} \) |
| 53 | \( 1 + 7.85T + 53T^{2} \) |
| 59 | \( 1 - 1.14T + 59T^{2} \) |
| 61 | \( 1 + 10.9T + 61T^{2} \) |
| 67 | \( 1 + 10.8T + 67T^{2} \) |
| 71 | \( 1 - 10.4T + 71T^{2} \) |
| 73 | \( 1 - 7.70T + 73T^{2} \) |
| 79 | \( 1 - 7.52T + 79T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 - 4.85T + 89T^{2} \) |
| 97 | \( 1 + 7.52T + 97T^{2} \) |
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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
−7.72523634000632720572484590598, −6.65555526162907147247212214304, −6.19923108327823659513937373247, −6.01441610941437545266970421446, −5.02187629087010750177594029915, −4.63091493511348701507694025635, −3.31634582528093141760363312500, −2.73328489930070959767425870474, −1.97373270537036396784146950833, −0.906268584465453014272905376262,
0.906268584465453014272905376262, 1.97373270537036396784146950833, 2.73328489930070959767425870474, 3.31634582528093141760363312500, 4.63091493511348701507694025635, 5.02187629087010750177594029915, 6.01441610941437545266970421446, 6.19923108327823659513937373247, 6.65555526162907147247212214304, 7.72523634000632720572484590598
