| L(s) = 1 | − 1.15·2-s − 6.67·4-s − 21.2·5-s + 31.2·7-s + 16.8·8-s + 24.3·10-s − 17.3·11-s − 35.9·14-s + 33.9·16-s + 89.1·17-s + 80.6·19-s + 141.·20-s + 19.9·22-s − 149.·23-s + 324.·25-s − 208.·28-s + 6.30·29-s − 78.3·31-s − 174.·32-s − 102.·34-s − 662.·35-s − 39.2·37-s − 92.7·38-s − 358.·40-s − 330.·41-s − 198.·43-s + 116.·44-s + ⋯ |
| L(s) = 1 | − 0.406·2-s − 0.834·4-s − 1.89·5-s + 1.68·7-s + 0.746·8-s + 0.771·10-s − 0.476·11-s − 0.686·14-s + 0.530·16-s + 1.27·17-s + 0.973·19-s + 1.58·20-s + 0.193·22-s − 1.35·23-s + 2.59·25-s − 1.40·28-s + 0.0403·29-s − 0.453·31-s − 0.962·32-s − 0.517·34-s − 3.20·35-s − 0.174·37-s − 0.396·38-s − 1.41·40-s − 1.25·41-s − 0.705·43-s + 0.397·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9151279636\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9151279636\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.15T + 8T^{2} \) |
| 5 | \( 1 + 21.2T + 125T^{2} \) |
| 7 | \( 1 - 31.2T + 343T^{2} \) |
| 11 | \( 1 + 17.3T + 1.33e3T^{2} \) |
| 17 | \( 1 - 89.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 80.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + 149.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 6.30T + 2.43e4T^{2} \) |
| 31 | \( 1 + 78.3T + 2.97e4T^{2} \) |
| 37 | \( 1 + 39.2T + 5.06e4T^{2} \) |
| 41 | \( 1 + 330.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 198.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 246.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 600.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 709.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 472.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 331.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 472.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 651.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 240.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 538.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 673.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 468.T + 9.12e5T^{2} \) |
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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
−8.773463284473765815540060517626, −8.023901326595047135396326894678, −7.929513004941529844926637986917, −7.22288396881798255485147880595, −5.42917335573588468249505163748, −4.86634970536439546380737676817, −4.07232071684357207105520598190, −3.32466831382516436736078327878, −1.54222504621604126805513965147, −0.53837408770501367720972753086,
0.53837408770501367720972753086, 1.54222504621604126805513965147, 3.32466831382516436736078327878, 4.07232071684357207105520598190, 4.86634970536439546380737676817, 5.42917335573588468249505163748, 7.22288396881798255485147880595, 7.929513004941529844926637986917, 8.023901326595047135396326894678, 8.773463284473765815540060517626
