| L(s) = 1 | − 0.300·2-s + 2.68·3-s − 1.90·4-s + 5-s − 0.806·6-s + 7-s + 1.17·8-s + 4.18·9-s − 0.300·10-s − 5.11·12-s + 3.71·13-s − 0.300·14-s + 2.68·15-s + 3.46·16-s + 2.57·17-s − 1.25·18-s + 4.28·19-s − 1.90·20-s + 2.68·21-s − 6.57·23-s + 3.15·24-s + 25-s − 1.11·26-s + 3.18·27-s − 1.90·28-s − 5.39·29-s − 0.806·30-s + ⋯ |
| L(s) = 1 | − 0.212·2-s + 1.54·3-s − 0.954·4-s + 0.447·5-s − 0.329·6-s + 0.377·7-s + 0.415·8-s + 1.39·9-s − 0.0950·10-s − 1.47·12-s + 1.02·13-s − 0.0803·14-s + 0.692·15-s + 0.866·16-s + 0.625·17-s − 0.296·18-s + 0.982·19-s − 0.426·20-s + 0.585·21-s − 1.37·23-s + 0.643·24-s + 0.200·25-s − 0.218·26-s + 0.613·27-s − 0.360·28-s − 1.00·29-s − 0.147·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.138228744\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.138228744\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 0.300T + 2T^{2} \) |
| 3 | \( 1 - 2.68T + 3T^{2} \) |
| 13 | \( 1 - 3.71T + 13T^{2} \) |
| 17 | \( 1 - 2.57T + 17T^{2} \) |
| 19 | \( 1 - 4.28T + 19T^{2} \) |
| 23 | \( 1 + 6.57T + 23T^{2} \) |
| 29 | \( 1 + 5.39T + 29T^{2} \) |
| 31 | \( 1 - 0.259T + 31T^{2} \) |
| 37 | \( 1 - 8.65T + 37T^{2} \) |
| 41 | \( 1 - 5.56T + 41T^{2} \) |
| 43 | \( 1 - 7.87T + 43T^{2} \) |
| 47 | \( 1 + 7.41T + 47T^{2} \) |
| 53 | \( 1 + 3.03T + 53T^{2} \) |
| 59 | \( 1 - 9.55T + 59T^{2} \) |
| 61 | \( 1 - 4.10T + 61T^{2} \) |
| 67 | \( 1 + 12.5T + 67T^{2} \) |
| 71 | \( 1 + 5.34T + 71T^{2} \) |
| 73 | \( 1 - 3.79T + 73T^{2} \) |
| 79 | \( 1 - 16.1T + 79T^{2} \) |
| 83 | \( 1 + 6.53T + 83T^{2} \) |
| 89 | \( 1 + 14.7T + 89T^{2} \) |
| 97 | \( 1 + 3.69T + 97T^{2} \) |
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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
−8.392105836098653452124700643592, −7.889567537012139058409017010199, −7.42423075094665089102692308099, −6.06695163319900245028378072040, −5.45726180265704193518968949862, −4.33326573622193911493919341108, −3.78711550826253836815682363356, −3.00237153147042020850338460233, −1.94140476818375211696385176731, −1.05406553550258667677469336917,
1.05406553550258667677469336917, 1.94140476818375211696385176731, 3.00237153147042020850338460233, 3.78711550826253836815682363356, 4.33326573622193911493919341108, 5.45726180265704193518968949862, 6.06695163319900245028378072040, 7.42423075094665089102692308099, 7.889567537012139058409017010199, 8.392105836098653452124700643592
