| L(s) = 1 | + 8.45·2-s − 3.68·3-s + 39.5·4-s − 25·5-s − 31.2·6-s + 45.3·7-s + 63.9·8-s − 229.·9-s − 211.·10-s + 121·11-s − 145.·12-s − 1.11e3·13-s + 383.·14-s + 92.2·15-s − 724.·16-s + 1.64e3·17-s − 1.94e3·18-s − 361·19-s − 989.·20-s − 167.·21-s + 1.02e3·22-s + 1.62e3·23-s − 236.·24-s + 625·25-s − 9.45e3·26-s + 1.74e3·27-s + 1.79e3·28-s + ⋯ |
| L(s) = 1 | + 1.49·2-s − 0.236·3-s + 1.23·4-s − 0.447·5-s − 0.353·6-s + 0.349·7-s + 0.353·8-s − 0.944·9-s − 0.668·10-s + 0.301·11-s − 0.292·12-s − 1.83·13-s + 0.522·14-s + 0.105·15-s − 0.707·16-s + 1.38·17-s − 1.41·18-s − 0.229·19-s − 0.552·20-s − 0.0826·21-s + 0.450·22-s + 0.641·23-s − 0.0836·24-s + 0.200·25-s − 2.74·26-s + 0.459·27-s + 0.432·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.467770347\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.467770347\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 25T \) |
| 11 | \( 1 - 121T \) |
| 19 | \( 1 + 361T \) |
| good | 2 | \( 1 - 8.45T + 32T^{2} \) |
| 3 | \( 1 + 3.68T + 243T^{2} \) |
| 7 | \( 1 - 45.3T + 1.68e4T^{2} \) |
| 13 | \( 1 + 1.11e3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.64e3T + 1.41e6T^{2} \) |
| 23 | \( 1 - 1.62e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.27e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 1.29e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 9.55e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 3.66e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.27e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 3.83e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 9.62e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.54e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.69e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.24e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.24e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.11e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.89e3T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.55e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 2.33e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.39e5T + 8.58e9T^{2} \) |
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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
−9.186315094893152503275807022158, −8.174381886966470888462909699691, −7.26057836385166827768086881468, −6.47448521573040041674947061443, −5.25193738325185126140353600728, −5.15250219430516277347030686864, −3.99764793194655960527363248964, −3.08485696445138549513642117575, −2.29743240489677241227950627817, −0.62081571347118692570854893850,
0.62081571347118692570854893850, 2.29743240489677241227950627817, 3.08485696445138549513642117575, 3.99764793194655960527363248964, 5.15250219430516277347030686864, 5.25193738325185126140353600728, 6.47448521573040041674947061443, 7.26057836385166827768086881468, 8.174381886966470888462909699691, 9.186315094893152503275807022158
