| L(s) = 1 | + 1.39·2-s + 2.10·3-s − 0.0457·4-s + 3.90·5-s + 2.94·6-s − 1.82·7-s − 2.85·8-s + 1.43·9-s + 5.46·10-s + 4.06·11-s − 0.0963·12-s − 2.56·13-s − 2.54·14-s + 8.22·15-s − 3.90·16-s + 2.96·17-s + 2.00·18-s − 3.33·19-s − 0.178·20-s − 3.83·21-s + 5.68·22-s − 9.09·23-s − 6.02·24-s + 10.2·25-s − 3.58·26-s − 3.30·27-s + 0.0834·28-s + ⋯ |
| L(s) = 1 | + 0.988·2-s + 1.21·3-s − 0.0228·4-s + 1.74·5-s + 1.20·6-s − 0.688·7-s − 1.01·8-s + 0.477·9-s + 1.72·10-s + 1.22·11-s − 0.0278·12-s − 0.712·13-s − 0.680·14-s + 2.12·15-s − 0.976·16-s + 0.717·17-s + 0.471·18-s − 0.764·19-s − 0.0400·20-s − 0.837·21-s + 1.21·22-s − 1.89·23-s − 1.22·24-s + 2.05·25-s − 0.703·26-s − 0.635·27-s + 0.0157·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 619 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 619 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.679663975\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.679663975\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 619 | \( 1 - T \) |
| good | 2 | \( 1 - 1.39T + 2T^{2} \) |
| 3 | \( 1 - 2.10T + 3T^{2} \) |
| 5 | \( 1 - 3.90T + 5T^{2} \) |
| 7 | \( 1 + 1.82T + 7T^{2} \) |
| 11 | \( 1 - 4.06T + 11T^{2} \) |
| 13 | \( 1 + 2.56T + 13T^{2} \) |
| 17 | \( 1 - 2.96T + 17T^{2} \) |
| 19 | \( 1 + 3.33T + 19T^{2} \) |
| 23 | \( 1 + 9.09T + 23T^{2} \) |
| 29 | \( 1 - 7.13T + 29T^{2} \) |
| 31 | \( 1 - 3.95T + 31T^{2} \) |
| 37 | \( 1 + 2.67T + 37T^{2} \) |
| 41 | \( 1 + 9.73T + 41T^{2} \) |
| 43 | \( 1 - 3.74T + 43T^{2} \) |
| 47 | \( 1 + 6.42T + 47T^{2} \) |
| 53 | \( 1 - 1.59T + 53T^{2} \) |
| 59 | \( 1 + 11.4T + 59T^{2} \) |
| 61 | \( 1 - 11.9T + 61T^{2} \) |
| 67 | \( 1 + 1.47T + 67T^{2} \) |
| 71 | \( 1 - 15.1T + 71T^{2} \) |
| 73 | \( 1 + 7.51T + 73T^{2} \) |
| 79 | \( 1 - 1.32T + 79T^{2} \) |
| 83 | \( 1 - 9.23T + 83T^{2} \) |
| 89 | \( 1 - 6.11T + 89T^{2} \) |
| 97 | \( 1 - 14.0T + 97T^{2} \) |
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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
−10.09480778275744326765761596793, −9.759803290381886199502616543777, −9.030477530516247313921340948734, −8.239489304088623233535091435336, −6.55102802903778658139380057832, −6.16975324151510698743937464973, −5.03630715887532020112064597445, −3.85051250312354486238210345980, −2.91317086754811013460864324720, −1.96485365607532540361497522307,
1.96485365607532540361497522307, 2.91317086754811013460864324720, 3.85051250312354486238210345980, 5.03630715887532020112064597445, 6.16975324151510698743937464973, 6.55102802903778658139380057832, 8.239489304088623233535091435336, 9.030477530516247313921340948734, 9.759803290381886199502616543777, 10.09480778275744326765761596793
