| L(s) = 1 | − 2-s + 4-s − 0.194·5-s − 1.95·7-s − 8-s + 0.194·10-s + 6.30·11-s + 4.81·13-s + 1.95·14-s + 16-s − 3.20·17-s − 0.859·19-s − 0.194·20-s − 6.30·22-s − 4.96·25-s − 4.81·26-s − 1.95·28-s − 5.41·29-s − 1.01·31-s − 32-s + 3.20·34-s + 0.379·35-s − 6.89·37-s + 0.859·38-s + 0.194·40-s + 0.972·41-s + 5.14·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.0868·5-s − 0.737·7-s − 0.353·8-s + 0.0614·10-s + 1.90·11-s + 1.33·13-s + 0.521·14-s + 0.250·16-s − 0.778·17-s − 0.197·19-s − 0.0434·20-s − 1.34·22-s − 0.992·25-s − 0.943·26-s − 0.368·28-s − 1.00·29-s − 0.182·31-s − 0.176·32-s + 0.550·34-s + 0.0640·35-s − 1.13·37-s + 0.139·38-s + 0.0307·40-s + 0.151·41-s + 0.784·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 + 0.194T + 5T^{2} \) |
| 7 | \( 1 + 1.95T + 7T^{2} \) |
| 11 | \( 1 - 6.30T + 11T^{2} \) |
| 13 | \( 1 - 4.81T + 13T^{2} \) |
| 17 | \( 1 + 3.20T + 17T^{2} \) |
| 19 | \( 1 + 0.859T + 19T^{2} \) |
| 29 | \( 1 + 5.41T + 29T^{2} \) |
| 31 | \( 1 + 1.01T + 31T^{2} \) |
| 37 | \( 1 + 6.89T + 37T^{2} \) |
| 41 | \( 1 - 0.972T + 41T^{2} \) |
| 43 | \( 1 - 5.14T + 43T^{2} \) |
| 47 | \( 1 - 7.41T + 47T^{2} \) |
| 53 | \( 1 - 2.00T + 53T^{2} \) |
| 59 | \( 1 - 8.60T + 59T^{2} \) |
| 61 | \( 1 + 12.1T + 61T^{2} \) |
| 67 | \( 1 + 7.09T + 67T^{2} \) |
| 71 | \( 1 + 1.10T + 71T^{2} \) |
| 73 | \( 1 + 13.1T + 73T^{2} \) |
| 79 | \( 1 + 1.12T + 79T^{2} \) |
| 83 | \( 1 - 4.00T + 83T^{2} \) |
| 89 | \( 1 + 9.55T + 89T^{2} \) |
| 97 | \( 1 - 1.12T + 97T^{2} \) |
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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
−7.20937918598312582208845850194, −6.77633666900788996408678732549, −6.10061933760953773324664033979, −5.68435194879367594089343452014, −4.18038671491398998348118665348, −3.89509272239103091689145221954, −3.06237239548002688242004080867, −1.89122699032836824168205236861, −1.23059218594296211352926861078, 0,
1.23059218594296211352926861078, 1.89122699032836824168205236861, 3.06237239548002688242004080867, 3.89509272239103091689145221954, 4.18038671491398998348118665348, 5.68435194879367594089343452014, 6.10061933760953773324664033979, 6.77633666900788996408678732549, 7.20937918598312582208845850194
