| L(s) = 1 | − 2.12·2-s − 3.24·3-s + 2.50·4-s − 1.62·5-s + 6.88·6-s + 1.55·7-s − 1.08·8-s + 7.50·9-s + 3.44·10-s + 2.86·11-s − 8.13·12-s − 2.30·13-s − 3.30·14-s + 5.25·15-s − 2.72·16-s + 5.43·17-s − 15.9·18-s + 1.20·19-s − 4.07·20-s − 5.04·21-s − 6.08·22-s − 5.07·23-s + 3.50·24-s − 2.36·25-s + 4.90·26-s − 14.5·27-s + 3.90·28-s + ⋯ |
| L(s) = 1 | − 1.50·2-s − 1.87·3-s + 1.25·4-s − 0.725·5-s + 2.80·6-s + 0.588·7-s − 0.382·8-s + 2.50·9-s + 1.09·10-s + 0.864·11-s − 2.34·12-s − 0.640·13-s − 0.883·14-s + 1.35·15-s − 0.680·16-s + 1.31·17-s − 3.75·18-s + 0.276·19-s − 0.910·20-s − 1.10·21-s − 1.29·22-s − 1.05·23-s + 0.715·24-s − 0.473·25-s + 0.961·26-s − 2.80·27-s + 0.738·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3371081047\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3371081047\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 71 | \( 1 - T \) |
| 113 | \( 1 + T \) |
| good | 2 | \( 1 + 2.12T + 2T^{2} \) |
| 3 | \( 1 + 3.24T + 3T^{2} \) |
| 5 | \( 1 + 1.62T + 5T^{2} \) |
| 7 | \( 1 - 1.55T + 7T^{2} \) |
| 11 | \( 1 - 2.86T + 11T^{2} \) |
| 13 | \( 1 + 2.30T + 13T^{2} \) |
| 17 | \( 1 - 5.43T + 17T^{2} \) |
| 19 | \( 1 - 1.20T + 19T^{2} \) |
| 23 | \( 1 + 5.07T + 23T^{2} \) |
| 29 | \( 1 + 7.51T + 29T^{2} \) |
| 31 | \( 1 - 0.692T + 31T^{2} \) |
| 37 | \( 1 - 6.78T + 37T^{2} \) |
| 41 | \( 1 + 1.50T + 41T^{2} \) |
| 43 | \( 1 + 1.78T + 43T^{2} \) |
| 47 | \( 1 - 0.490T + 47T^{2} \) |
| 53 | \( 1 - 12.5T + 53T^{2} \) |
| 59 | \( 1 - 7.12T + 59T^{2} \) |
| 61 | \( 1 + 1.38T + 61T^{2} \) |
| 67 | \( 1 - 7.74T + 67T^{2} \) |
| 73 | \( 1 - 12.6T + 73T^{2} \) |
| 79 | \( 1 - 2.52T + 79T^{2} \) |
| 83 | \( 1 + 12.4T + 83T^{2} \) |
| 89 | \( 1 + 13.9T + 89T^{2} \) |
| 97 | \( 1 - 14.2T + 97T^{2} \) |
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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
−7.76073826757487956433970776107, −7.28418522953307271112537432193, −6.67446934880850889058952309457, −5.82047779500617414934382346473, −5.25853334764402512845104398842, −4.34492729890603398172261953876, −3.77931907986419932170470423865, −2.03906527797524504524695526053, −1.22641630038309791778688002864, −0.47270386015272579471446645489,
0.47270386015272579471446645489, 1.22641630038309791778688002864, 2.03906527797524504524695526053, 3.77931907986419932170470423865, 4.34492729890603398172261953876, 5.25853334764402512845104398842, 5.82047779500617414934382346473, 6.67446934880850889058952309457, 7.28418522953307271112537432193, 7.76073826757487956433970776107
