| L(s) = 1 | + 1.46·2-s + 0.139·4-s − 2.39·5-s + 7-s − 2.72·8-s − 3.50·10-s − 5.04·13-s + 1.46·14-s − 4.25·16-s − 6.36·17-s + 5.32·19-s − 0.333·20-s − 4.92·23-s + 0.751·25-s − 7.37·26-s + 0.139·28-s + 5.04·29-s − 7.57·31-s − 0.786·32-s − 9.31·34-s − 2.39·35-s + 4.24·37-s + 7.78·38-s + 6.52·40-s − 0.646·41-s + 10.5·43-s − 7.20·46-s + ⋯ |
| L(s) = 1 | + 1.03·2-s + 0.0695·4-s − 1.07·5-s + 0.377·7-s − 0.962·8-s − 1.10·10-s − 1.39·13-s + 0.390·14-s − 1.06·16-s − 1.54·17-s + 1.22·19-s − 0.0746·20-s − 1.02·23-s + 0.150·25-s − 1.44·26-s + 0.0263·28-s + 0.936·29-s − 1.35·31-s − 0.138·32-s − 1.59·34-s − 0.405·35-s + 0.698·37-s + 1.26·38-s + 1.03·40-s − 0.101·41-s + 1.60·43-s − 1.06·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.322058887\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.322058887\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 1.46T + 2T^{2} \) |
| 5 | \( 1 + 2.39T + 5T^{2} \) |
| 13 | \( 1 + 5.04T + 13T^{2} \) |
| 17 | \( 1 + 6.36T + 17T^{2} \) |
| 19 | \( 1 - 5.32T + 19T^{2} \) |
| 23 | \( 1 + 4.92T + 23T^{2} \) |
| 29 | \( 1 - 5.04T + 29T^{2} \) |
| 31 | \( 1 + 7.57T + 31T^{2} \) |
| 37 | \( 1 - 4.24T + 37T^{2} \) |
| 41 | \( 1 + 0.646T + 41T^{2} \) |
| 43 | \( 1 - 10.5T + 43T^{2} \) |
| 47 | \( 1 + 0.526T + 47T^{2} \) |
| 53 | \( 1 + 3.72T + 53T^{2} \) |
| 59 | \( 1 + 7.97T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 8.76T + 67T^{2} \) |
| 71 | \( 1 - 11.4T + 71T^{2} \) |
| 73 | \( 1 - 13.0T + 73T^{2} \) |
| 79 | \( 1 + 11.4T + 79T^{2} \) |
| 83 | \( 1 - 13.1T + 83T^{2} \) |
| 89 | \( 1 + 11.8T + 89T^{2} \) |
| 97 | \( 1 + 1.87T + 97T^{2} \) |
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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
−7.79461256769360828674849544806, −7.16074104159883259390767368823, −6.43324305533865202982805985774, −5.51872664774747092720448766797, −4.91779128438094852679397298397, −4.29525487019604110628010507227, −3.82648759130211974008494397741, −2.87278688441794125276318957341, −2.13709266336475789557991943183, −0.46494875627773215504581136371,
0.46494875627773215504581136371, 2.13709266336475789557991943183, 2.87278688441794125276318957341, 3.82648759130211974008494397741, 4.29525487019604110628010507227, 4.91779128438094852679397298397, 5.51872664774747092720448766797, 6.43324305533865202982805985774, 7.16074104159883259390767368823, 7.79461256769360828674849544806
