| L(s) = 1 | − 1.91·2-s + 1.67·4-s + 5-s + 4.35·7-s + 0.619·8-s − 1.91·10-s − 11-s + 4.48·13-s − 8.35·14-s − 4.54·16-s − 2.68·17-s + 19-s + 1.67·20-s + 1.91·22-s + 2.22·23-s + 25-s − 8.60·26-s + 7.30·28-s + 8.34·29-s + 4.48·31-s + 7.47·32-s + 5.15·34-s + 4.35·35-s − 10.6·37-s − 1.91·38-s + 0.619·40-s − 0.241·41-s + ⋯ |
| L(s) = 1 | − 1.35·2-s + 0.838·4-s + 0.447·5-s + 1.64·7-s + 0.218·8-s − 0.606·10-s − 0.301·11-s + 1.24·13-s − 2.23·14-s − 1.13·16-s − 0.651·17-s + 0.229·19-s + 0.374·20-s + 0.408·22-s + 0.462·23-s + 0.200·25-s − 1.68·26-s + 1.38·28-s + 1.54·29-s + 0.805·31-s + 1.32·32-s + 0.883·34-s + 0.736·35-s − 1.75·37-s − 0.311·38-s + 0.0979·40-s − 0.0377·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.555730634\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.555730634\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + 1.91T + 2T^{2} \) |
| 7 | \( 1 - 4.35T + 7T^{2} \) |
| 13 | \( 1 - 4.48T + 13T^{2} \) |
| 17 | \( 1 + 2.68T + 17T^{2} \) |
| 23 | \( 1 - 2.22T + 23T^{2} \) |
| 29 | \( 1 - 8.34T + 29T^{2} \) |
| 31 | \( 1 - 4.48T + 31T^{2} \) |
| 37 | \( 1 + 10.6T + 37T^{2} \) |
| 41 | \( 1 + 0.241T + 41T^{2} \) |
| 43 | \( 1 - 12.4T + 43T^{2} \) |
| 47 | \( 1 + 6.41T + 47T^{2} \) |
| 53 | \( 1 + 4.35T + 53T^{2} \) |
| 59 | \( 1 + 7.32T + 59T^{2} \) |
| 61 | \( 1 + 7.45T + 61T^{2} \) |
| 67 | \( 1 + 12.5T + 67T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 + 2.70T + 73T^{2} \) |
| 79 | \( 1 - 8.67T + 79T^{2} \) |
| 83 | \( 1 + 1.01T + 83T^{2} \) |
| 89 | \( 1 + 11.1T + 89T^{2} \) |
| 97 | \( 1 - 17.2T + 97T^{2} \) |
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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
−7.919522022719882087506816479542, −7.29633100463934568892165368008, −6.51931449423514764670570934277, −5.79208530029460889378745276595, −4.71674556123410042423682298674, −4.57969072958510199606937155632, −3.21264753785828266711070887684, −2.15166436946656493550309472268, −1.50816533933860252444864754025, −0.815988167783188965205989154198,
0.815988167783188965205989154198, 1.50816533933860252444864754025, 2.15166436946656493550309472268, 3.21264753785828266711070887684, 4.57969072958510199606937155632, 4.71674556123410042423682298674, 5.79208530029460889378745276595, 6.51931449423514764670570934277, 7.29633100463934568892165368008, 7.919522022719882087506816479542
