| L(s) = 1 | − 1.52·2-s + 0.315·4-s + 3.19·5-s + 1.20·7-s + 2.56·8-s − 4.86·10-s − 11-s + 2.29·13-s − 1.83·14-s − 4.53·16-s − 3.92·17-s + 3.12·19-s + 1.00·20-s + 1.52·22-s + 0.732·23-s + 5.20·25-s − 3.48·26-s + 0.379·28-s − 0.00873·29-s + 4.48·31-s + 1.76·32-s + 5.97·34-s + 3.84·35-s + 3.89·37-s − 4.75·38-s + 8.18·40-s − 3.44·41-s + ⋯ |
| L(s) = 1 | − 1.07·2-s + 0.157·4-s + 1.42·5-s + 0.454·7-s + 0.906·8-s − 1.53·10-s − 0.301·11-s + 0.635·13-s − 0.489·14-s − 1.13·16-s − 0.953·17-s + 0.716·19-s + 0.225·20-s + 0.324·22-s + 0.152·23-s + 1.04·25-s − 0.683·26-s + 0.0717·28-s − 0.00162·29-s + 0.804·31-s + 0.312·32-s + 1.02·34-s + 0.649·35-s + 0.640·37-s − 0.770·38-s + 1.29·40-s − 0.538·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6633 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6633 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.526057324\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.526057324\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 67 | \( 1 + T \) |
| good | 2 | \( 1 + 1.52T + 2T^{2} \) |
| 5 | \( 1 - 3.19T + 5T^{2} \) |
| 7 | \( 1 - 1.20T + 7T^{2} \) |
| 13 | \( 1 - 2.29T + 13T^{2} \) |
| 17 | \( 1 + 3.92T + 17T^{2} \) |
| 19 | \( 1 - 3.12T + 19T^{2} \) |
| 23 | \( 1 - 0.732T + 23T^{2} \) |
| 29 | \( 1 + 0.00873T + 29T^{2} \) |
| 31 | \( 1 - 4.48T + 31T^{2} \) |
| 37 | \( 1 - 3.89T + 37T^{2} \) |
| 41 | \( 1 + 3.44T + 41T^{2} \) |
| 43 | \( 1 + 3.72T + 43T^{2} \) |
| 47 | \( 1 + 6.57T + 47T^{2} \) |
| 53 | \( 1 - 4.83T + 53T^{2} \) |
| 59 | \( 1 + 6.03T + 59T^{2} \) |
| 61 | \( 1 - 2.41T + 61T^{2} \) |
| 71 | \( 1 - 10.1T + 71T^{2} \) |
| 73 | \( 1 - 10.7T + 73T^{2} \) |
| 79 | \( 1 - 9.20T + 79T^{2} \) |
| 83 | \( 1 + 2.76T + 83T^{2} \) |
| 89 | \( 1 + 2.67T + 89T^{2} \) |
| 97 | \( 1 - 6.39T + 97T^{2} \) |
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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
−8.222044851537053820119349804267, −7.46071769482095266578689048464, −6.60405369150687167678140614445, −6.06727351819814183701281597625, −5.07223997737482703048406485514, −4.68749275116955831894926908287, −3.45087625694602107839831536057, −2.29712794282307597796465338485, −1.69855970879284028723834144603, −0.78828389496862962495320324681,
0.78828389496862962495320324681, 1.69855970879284028723834144603, 2.29712794282307597796465338485, 3.45087625694602107839831536057, 4.68749275116955831894926908287, 5.07223997737482703048406485514, 6.06727351819814183701281597625, 6.60405369150687167678140614445, 7.46071769482095266578689048464, 8.222044851537053820119349804267
