| L(s) = 1 | − 2.15·3-s − 0.0166·5-s − 4.51·7-s + 1.64·9-s + 1.00·11-s − 5.74·13-s + 0.0359·15-s − 6.04·17-s + 5.55·19-s + 9.72·21-s + 0.430·23-s − 4.99·25-s + 2.92·27-s + 0.293·29-s − 9.26·31-s − 2.15·33-s + 0.0752·35-s − 8.67·37-s + 12.3·39-s − 11.9·41-s − 5.90·43-s − 0.0274·45-s − 13.2·47-s + 13.3·49-s + 13.0·51-s − 0.840·53-s − 0.0166·55-s + ⋯ |
| L(s) = 1 | − 1.24·3-s − 0.00746·5-s − 1.70·7-s + 0.547·9-s + 0.301·11-s − 1.59·13-s + 0.00928·15-s − 1.46·17-s + 1.27·19-s + 2.12·21-s + 0.0896·23-s − 0.999·25-s + 0.562·27-s + 0.0544·29-s − 1.66·31-s − 0.375·33-s + 0.0127·35-s − 1.42·37-s + 1.98·39-s − 1.86·41-s − 0.900·43-s − 0.00408·45-s − 1.92·47-s + 1.90·49-s + 1.82·51-s − 0.115·53-s − 0.00225·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.07897091581\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.07897091581\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 503 | \( 1 - T \) |
| good | 3 | \( 1 + 2.15T + 3T^{2} \) |
| 5 | \( 1 + 0.0166T + 5T^{2} \) |
| 7 | \( 1 + 4.51T + 7T^{2} \) |
| 11 | \( 1 - 1.00T + 11T^{2} \) |
| 13 | \( 1 + 5.74T + 13T^{2} \) |
| 17 | \( 1 + 6.04T + 17T^{2} \) |
| 19 | \( 1 - 5.55T + 19T^{2} \) |
| 23 | \( 1 - 0.430T + 23T^{2} \) |
| 29 | \( 1 - 0.293T + 29T^{2} \) |
| 31 | \( 1 + 9.26T + 31T^{2} \) |
| 37 | \( 1 + 8.67T + 37T^{2} \) |
| 41 | \( 1 + 11.9T + 41T^{2} \) |
| 43 | \( 1 + 5.90T + 43T^{2} \) |
| 47 | \( 1 + 13.2T + 47T^{2} \) |
| 53 | \( 1 + 0.840T + 53T^{2} \) |
| 59 | \( 1 - 13.3T + 59T^{2} \) |
| 61 | \( 1 + 3.84T + 61T^{2} \) |
| 67 | \( 1 - 3.29T + 67T^{2} \) |
| 71 | \( 1 - 1.77T + 71T^{2} \) |
| 73 | \( 1 + 1.32T + 73T^{2} \) |
| 79 | \( 1 - 15.9T + 79T^{2} \) |
| 83 | \( 1 + 14.8T + 83T^{2} \) |
| 89 | \( 1 + 4.18T + 89T^{2} \) |
| 97 | \( 1 - 17.1T + 97T^{2} \) |
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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
−8.556060048196442119587845061429, −7.30917765053623707813245592497, −6.80779428426409910765837543587, −6.39531421175885247466772330747, −5.34565852315824129589760501755, −5.04331246756430666797260682982, −3.81237585519212577154984798016, −3.06913550073986610586586987670, −1.90978146843317255607220157680, −0.16007299712592972911390615819,
0.16007299712592972911390615819, 1.90978146843317255607220157680, 3.06913550073986610586586987670, 3.81237585519212577154984798016, 5.04331246756430666797260682982, 5.34565852315824129589760501755, 6.39531421175885247466772330747, 6.80779428426409910765837543587, 7.30917765053623707813245592497, 8.556060048196442119587845061429
