| L(s) = 1 | + 1.13·2-s + 1.10·3-s − 0.714·4-s + 1.25·6-s − 2.46·7-s − 3.07·8-s − 1.77·9-s + 1.71·11-s − 0.790·12-s − 6.49·13-s − 2.79·14-s − 2.05·16-s − 3.32·17-s − 2.01·18-s + 0.734·19-s − 2.72·21-s + 1.94·22-s + 2.08·23-s − 3.40·24-s − 7.35·26-s − 5.28·27-s + 1.75·28-s − 4.21·29-s + 7.46·31-s + 3.82·32-s + 1.89·33-s − 3.77·34-s + ⋯ |
| L(s) = 1 | + 0.801·2-s + 0.638·3-s − 0.357·4-s + 0.511·6-s − 0.930·7-s − 1.08·8-s − 0.592·9-s + 0.517·11-s − 0.228·12-s − 1.80·13-s − 0.745·14-s − 0.514·16-s − 0.807·17-s − 0.474·18-s + 0.168·19-s − 0.593·21-s + 0.414·22-s + 0.434·23-s − 0.694·24-s − 1.44·26-s − 1.01·27-s + 0.332·28-s − 0.782·29-s + 1.34·31-s + 0.675·32-s + 0.330·33-s − 0.647·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 2 | \( 1 - 1.13T + 2T^{2} \) |
| 3 | \( 1 - 1.10T + 3T^{2} \) |
| 7 | \( 1 + 2.46T + 7T^{2} \) |
| 11 | \( 1 - 1.71T + 11T^{2} \) |
| 13 | \( 1 + 6.49T + 13T^{2} \) |
| 17 | \( 1 + 3.32T + 17T^{2} \) |
| 19 | \( 1 - 0.734T + 19T^{2} \) |
| 23 | \( 1 - 2.08T + 23T^{2} \) |
| 29 | \( 1 + 4.21T + 29T^{2} \) |
| 31 | \( 1 - 7.46T + 31T^{2} \) |
| 41 | \( 1 - 1.71T + 41T^{2} \) |
| 43 | \( 1 + 1.81T + 43T^{2} \) |
| 47 | \( 1 - 0.882T + 47T^{2} \) |
| 53 | \( 1 - 7.03T + 53T^{2} \) |
| 59 | \( 1 - 0.387T + 59T^{2} \) |
| 61 | \( 1 + 11.8T + 61T^{2} \) |
| 67 | \( 1 + 12.1T + 67T^{2} \) |
| 71 | \( 1 - 13.7T + 71T^{2} \) |
| 73 | \( 1 + 16.6T + 73T^{2} \) |
| 79 | \( 1 - 8.23T + 79T^{2} \) |
| 83 | \( 1 - 4.80T + 83T^{2} \) |
| 89 | \( 1 + 1.52T + 89T^{2} \) |
| 97 | \( 1 - 18.1T + 97T^{2} \) |
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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
−9.324966014963493823427466479406, −9.153984281060148643092278915416, −8.015983741771727049645058738129, −6.96126613518207108477182675256, −6.10209691831214886062966583163, −5.10996007972715214353682137927, −4.22586525859944265601599559748, −3.18510171781422560197161661049, −2.49761115346100632138825496932, 0,
2.49761115346100632138825496932, 3.18510171781422560197161661049, 4.22586525859944265601599559748, 5.10996007972715214353682137927, 6.10209691831214886062966583163, 6.96126613518207108477182675256, 8.015983741771727049645058738129, 9.153984281060148643092278915416, 9.324966014963493823427466479406
