| L(s) = 1 | + 1.06·2-s − 0.862·4-s − 3.84·5-s − 1.30·7-s − 3.05·8-s − 4.10·10-s − 11-s + 1.88·13-s − 1.39·14-s − 1.53·16-s + 6.08·17-s + 5.65·19-s + 3.31·20-s − 1.06·22-s − 3.88·23-s + 9.79·25-s + 2.01·26-s + 1.12·28-s − 5.21·29-s + 4.43·31-s + 4.47·32-s + 6.48·34-s + 5.02·35-s + 0.333·37-s + 6.03·38-s + 11.7·40-s − 1.38·41-s + ⋯ |
| L(s) = 1 | + 0.754·2-s − 0.431·4-s − 1.72·5-s − 0.494·7-s − 1.07·8-s − 1.29·10-s − 0.301·11-s + 0.523·13-s − 0.372·14-s − 0.382·16-s + 1.47·17-s + 1.29·19-s + 0.741·20-s − 0.227·22-s − 0.810·23-s + 1.95·25-s + 0.394·26-s + 0.213·28-s − 0.968·29-s + 0.795·31-s + 0.790·32-s + 1.11·34-s + 0.849·35-s + 0.0548·37-s + 0.978·38-s + 1.85·40-s − 0.215·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 - 1.06T + 2T^{2} \) |
| 5 | \( 1 + 3.84T + 5T^{2} \) |
| 7 | \( 1 + 1.30T + 7T^{2} \) |
| 13 | \( 1 - 1.88T + 13T^{2} \) |
| 17 | \( 1 - 6.08T + 17T^{2} \) |
| 19 | \( 1 - 5.65T + 19T^{2} \) |
| 23 | \( 1 + 3.88T + 23T^{2} \) |
| 29 | \( 1 + 5.21T + 29T^{2} \) |
| 31 | \( 1 - 4.43T + 31T^{2} \) |
| 37 | \( 1 - 0.333T + 37T^{2} \) |
| 41 | \( 1 + 1.38T + 41T^{2} \) |
| 43 | \( 1 - 2.89T + 43T^{2} \) |
| 47 | \( 1 - 8.79T + 47T^{2} \) |
| 53 | \( 1 - 8.65T + 53T^{2} \) |
| 59 | \( 1 + 8.72T + 59T^{2} \) |
| 67 | \( 1 + 13.6T + 67T^{2} \) |
| 71 | \( 1 + 8.66T + 71T^{2} \) |
| 73 | \( 1 + 3.51T + 73T^{2} \) |
| 79 | \( 1 - 12.5T + 79T^{2} \) |
| 83 | \( 1 + 17.3T + 83T^{2} \) |
| 89 | \( 1 + 16.4T + 89T^{2} \) |
| 97 | \( 1 - 0.868T + 97T^{2} \) |
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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
−7.63822907956000202275862603741, −7.22046926243247656507804691306, −6.01533114478678344606014041965, −5.56642845932149154061931849065, −4.62994922432018344207729911266, −4.00529256928487830678957611590, −3.37285486993568785333844053937, −2.94562427930957243631164845340, −1.04924646397413899700909558597, 0,
1.04924646397413899700909558597, 2.94562427930957243631164845340, 3.37285486993568785333844053937, 4.00529256928487830678957611590, 4.62994922432018344207729911266, 5.56642845932149154061931849065, 6.01533114478678344606014041965, 7.22046926243247656507804691306, 7.63822907956000202275862603741
