| L(s) = 1 | + 1.87·2-s + 1.53·4-s − 0.120·5-s − 1.53·7-s − 0.879·8-s − 0.226·10-s − 2.69·11-s + 4.57·13-s − 2.87·14-s − 4.71·16-s − 1.87·19-s − 0.184·20-s − 5.06·22-s − 7.41·23-s − 4.98·25-s + 8.59·26-s − 2.34·28-s − 3.41·29-s + 3.83·31-s − 7.10·32-s + 0.184·35-s − 5.24·37-s − 3.53·38-s + 0.106·40-s − 6.24·41-s + 5.49·43-s − 4.12·44-s + ⋯ |
| L(s) = 1 | + 1.32·2-s + 0.766·4-s − 0.0539·5-s − 0.579·7-s − 0.310·8-s − 0.0716·10-s − 0.812·11-s + 1.26·13-s − 0.769·14-s − 1.17·16-s − 0.431·19-s − 0.0413·20-s − 1.07·22-s − 1.54·23-s − 0.997·25-s + 1.68·26-s − 0.443·28-s − 0.633·29-s + 0.689·31-s − 1.25·32-s + 0.0312·35-s − 0.862·37-s − 0.572·38-s + 0.0167·40-s − 0.975·41-s + 0.837·43-s − 0.622·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 1.87T + 2T^{2} \) |
| 5 | \( 1 + 0.120T + 5T^{2} \) |
| 7 | \( 1 + 1.53T + 7T^{2} \) |
| 11 | \( 1 + 2.69T + 11T^{2} \) |
| 13 | \( 1 - 4.57T + 13T^{2} \) |
| 19 | \( 1 + 1.87T + 19T^{2} \) |
| 23 | \( 1 + 7.41T + 23T^{2} \) |
| 29 | \( 1 + 3.41T + 29T^{2} \) |
| 31 | \( 1 - 3.83T + 31T^{2} \) |
| 37 | \( 1 + 5.24T + 37T^{2} \) |
| 41 | \( 1 + 6.24T + 41T^{2} \) |
| 43 | \( 1 - 5.49T + 43T^{2} \) |
| 47 | \( 1 - 7.34T + 47T^{2} \) |
| 53 | \( 1 + 0.822T + 53T^{2} \) |
| 59 | \( 1 - 3.14T + 59T^{2} \) |
| 61 | \( 1 - 1.22T + 61T^{2} \) |
| 67 | \( 1 + 14.6T + 67T^{2} \) |
| 71 | \( 1 - 0.170T + 71T^{2} \) |
| 73 | \( 1 + 11.8T + 73T^{2} \) |
| 79 | \( 1 + 4.14T + 79T^{2} \) |
| 83 | \( 1 - 2.43T + 83T^{2} \) |
| 89 | \( 1 - 15.0T + 89T^{2} \) |
| 97 | \( 1 - 11.1T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.443975573935946361547728975210, −7.64664430997919530625681563960, −6.58953722315549995687934727943, −5.97103558250766146029034892550, −5.47415572103556894114790791208, −4.35421620549798529336860977905, −3.78406056128829644051309521379, −2.98922568105921940236235900647, −1.93814225290713998147166399351, 0,
1.93814225290713998147166399351, 2.98922568105921940236235900647, 3.78406056128829644051309521379, 4.35421620549798529336860977905, 5.47415572103556894114790791208, 5.97103558250766146029034892550, 6.58953722315549995687934727943, 7.64664430997919530625681563960, 8.443975573935946361547728975210
