| L(s) = 1 | − 2.44·2-s − 0.121·3-s + 3.97·4-s + 0.297·6-s − 0.414·7-s − 4.82·8-s − 2.98·9-s + 4.59·11-s − 0.484·12-s + 1.01·14-s + 3.83·16-s − 0.744·17-s + 7.29·18-s + 7.47·19-s + 0.0504·21-s − 11.2·22-s − 6.35·23-s + 0.587·24-s + 0.729·27-s − 1.64·28-s − 5.60·29-s − 3.60·31-s + 0.261·32-s − 0.559·33-s + 1.81·34-s − 11.8·36-s − 6.43·37-s + ⋯ |
| L(s) = 1 | − 1.72·2-s − 0.0703·3-s + 1.98·4-s + 0.121·6-s − 0.156·7-s − 1.70·8-s − 0.995·9-s + 1.38·11-s − 0.139·12-s + 0.270·14-s + 0.959·16-s − 0.180·17-s + 1.71·18-s + 1.71·19-s + 0.0110·21-s − 2.39·22-s − 1.32·23-s + 0.119·24-s + 0.140·27-s − 0.310·28-s − 1.04·29-s − 0.648·31-s + 0.0462·32-s − 0.0973·33-s + 0.311·34-s − 1.97·36-s − 1.05·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 2.44T + 2T^{2} \) |
| 3 | \( 1 + 0.121T + 3T^{2} \) |
| 7 | \( 1 + 0.414T + 7T^{2} \) |
| 11 | \( 1 - 4.59T + 11T^{2} \) |
| 17 | \( 1 + 0.744T + 17T^{2} \) |
| 19 | \( 1 - 7.47T + 19T^{2} \) |
| 23 | \( 1 + 6.35T + 23T^{2} \) |
| 29 | \( 1 + 5.60T + 29T^{2} \) |
| 31 | \( 1 + 3.60T + 31T^{2} \) |
| 37 | \( 1 + 6.43T + 37T^{2} \) |
| 41 | \( 1 - 5.21T + 41T^{2} \) |
| 43 | \( 1 - 9.93T + 43T^{2} \) |
| 47 | \( 1 + 1.52T + 47T^{2} \) |
| 53 | \( 1 + 7.56T + 53T^{2} \) |
| 59 | \( 1 + 3.64T + 59T^{2} \) |
| 61 | \( 1 - 11.9T + 61T^{2} \) |
| 67 | \( 1 + 6.55T + 67T^{2} \) |
| 71 | \( 1 - 6.80T + 71T^{2} \) |
| 73 | \( 1 + 3.08T + 73T^{2} \) |
| 79 | \( 1 - 1.13T + 79T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 + 6.97T + 89T^{2} \) |
| 97 | \( 1 + 12.7T + 97T^{2} \) |
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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
−8.118396466591409456597290963073, −7.54117924309335746166635049154, −6.80333004171237927392131591766, −6.09229352743780639233143835100, −5.39180008455368363018727583503, −3.99138278884982194206456607858, −3.13209729082156866088053148966, −2.06244669072993441218344115854, −1.17218741054609020549727689975, 0,
1.17218741054609020549727689975, 2.06244669072993441218344115854, 3.13209729082156866088053148966, 3.99138278884982194206456607858, 5.39180008455368363018727583503, 6.09229352743780639233143835100, 6.80333004171237927392131591766, 7.54117924309335746166635049154, 8.118396466591409456597290963073
