| L(s) = 1 | + 2.10·2-s − 2.97·3-s + 2.41·4-s − 6.24·6-s − 4.82·7-s + 0.870·8-s + 5.82·9-s + 2·11-s − 7.17·12-s + 1.23·13-s − 10.1·14-s − 2.99·16-s + 3.65·17-s + 12.2·18-s + 14.3·21-s + 4.20·22-s − 4.82·23-s − 2.58·24-s + 2.58·26-s − 8.40·27-s − 11.6·28-s − 2.46·29-s + 5.94·31-s − 8.04·32-s − 5.94·33-s + 7.68·34-s + 14.0·36-s + ⋯ |
| L(s) = 1 | + 1.48·2-s − 1.71·3-s + 1.20·4-s − 2.54·6-s − 1.82·7-s + 0.307·8-s + 1.94·9-s + 0.603·11-s − 2.07·12-s + 0.341·13-s − 2.71·14-s − 0.749·16-s + 0.886·17-s + 2.88·18-s + 3.13·21-s + 0.895·22-s − 1.00·23-s − 0.527·24-s + 0.507·26-s − 1.61·27-s − 2.20·28-s − 0.457·29-s + 1.06·31-s − 1.42·32-s − 1.03·33-s + 1.31·34-s + 2.34·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - 2.10T + 2T^{2} \) |
| 3 | \( 1 + 2.97T + 3T^{2} \) |
| 7 | \( 1 + 4.82T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 1.23T + 13T^{2} \) |
| 17 | \( 1 - 3.65T + 17T^{2} \) |
| 23 | \( 1 + 4.82T + 23T^{2} \) |
| 29 | \( 1 + 2.46T + 29T^{2} \) |
| 31 | \( 1 - 5.94T + 31T^{2} \) |
| 37 | \( 1 - 7.17T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 0.828T + 43T^{2} \) |
| 47 | \( 1 - 6.48T + 47T^{2} \) |
| 53 | \( 1 - 4.71T + 53T^{2} \) |
| 59 | \( 1 + 11.8T + 59T^{2} \) |
| 61 | \( 1 - 2.82T + 61T^{2} \) |
| 67 | \( 1 - 5.43T + 67T^{2} \) |
| 71 | \( 1 + 10.8T + 71T^{2} \) |
| 73 | \( 1 - 0.343T + 73T^{2} \) |
| 79 | \( 1 - 11.8T + 79T^{2} \) |
| 83 | \( 1 + 3.17T + 83T^{2} \) |
| 89 | \( 1 + 9.42T + 89T^{2} \) |
| 97 | \( 1 - 1.23T + 97T^{2} \) |
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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
−6.86999680502596740118816385692, −6.24012274631180881886434200133, −6.04511350216289563412907496139, −5.52590116274528498389763865203, −4.62037667440260048112806831466, −3.98248720872125684332591481607, −3.43837781564133631463341469192, −2.51125621459489193678343230592, −1.06373579888924481848007889062, 0,
1.06373579888924481848007889062, 2.51125621459489193678343230592, 3.43837781564133631463341469192, 3.98248720872125684332591481607, 4.62037667440260048112806831466, 5.52590116274528498389763865203, 6.04511350216289563412907496139, 6.24012274631180881886434200133, 6.86999680502596740118816385692
