| L(s) = 1 | + 2.05·2-s + 2.23·4-s − 0.00207·5-s − 2.05·7-s + 0.481·8-s − 0.00427·10-s − 4.53·11-s + 7.11·13-s − 4.22·14-s − 3.47·16-s + 5.07·17-s − 19-s − 0.00464·20-s − 9.32·22-s + 4.14·23-s − 4.99·25-s + 14.6·26-s − 4.58·28-s + 1.14·29-s + 6.25·31-s − 8.11·32-s + 10.4·34-s + 0.00427·35-s − 0.0757·37-s − 2.05·38-s − 0.00100·40-s − 9.06·41-s + ⋯ |
| L(s) = 1 | + 1.45·2-s + 1.11·4-s − 0.000929·5-s − 0.776·7-s + 0.170·8-s − 0.00135·10-s − 1.36·11-s + 1.97·13-s − 1.12·14-s − 0.869·16-s + 1.23·17-s − 0.229·19-s − 0.00103·20-s − 1.98·22-s + 0.864·23-s − 0.999·25-s + 2.87·26-s − 0.867·28-s + 0.212·29-s + 1.12·31-s − 1.43·32-s + 1.79·34-s + 0.000722·35-s − 0.0124·37-s − 0.333·38-s − 0.000158·40-s − 1.41·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.063782035\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.063782035\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 + T \) |
| 47 | \( 1 - T \) |
| good | 2 | \( 1 - 2.05T + 2T^{2} \) |
| 5 | \( 1 + 0.00207T + 5T^{2} \) |
| 7 | \( 1 + 2.05T + 7T^{2} \) |
| 11 | \( 1 + 4.53T + 11T^{2} \) |
| 13 | \( 1 - 7.11T + 13T^{2} \) |
| 17 | \( 1 - 5.07T + 17T^{2} \) |
| 23 | \( 1 - 4.14T + 23T^{2} \) |
| 29 | \( 1 - 1.14T + 29T^{2} \) |
| 31 | \( 1 - 6.25T + 31T^{2} \) |
| 37 | \( 1 + 0.0757T + 37T^{2} \) |
| 41 | \( 1 + 9.06T + 41T^{2} \) |
| 43 | \( 1 - 6.84T + 43T^{2} \) |
| 53 | \( 1 - 9.99T + 53T^{2} \) |
| 59 | \( 1 - 9.70T + 59T^{2} \) |
| 61 | \( 1 + 6.52T + 61T^{2} \) |
| 67 | \( 1 - 11.0T + 67T^{2} \) |
| 71 | \( 1 + 4.38T + 71T^{2} \) |
| 73 | \( 1 + 9.37T + 73T^{2} \) |
| 79 | \( 1 - 1.34T + 79T^{2} \) |
| 83 | \( 1 - 15.8T + 83T^{2} \) |
| 89 | \( 1 - 11.4T + 89T^{2} \) |
| 97 | \( 1 + 2.70T + 97T^{2} \) |
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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
−7.72324807742982311876303502241, −6.82548206853928445309733859462, −6.17602596805243030125264376862, −5.69592348337304702412646216559, −5.11803002163668632979531790674, −4.22811156153174533081051037032, −3.43073818309742878040563403227, −3.13384934338031226048094025615, −2.15322010003677943006905731605, −0.78817290099880203565905814985,
0.78817290099880203565905814985, 2.15322010003677943006905731605, 3.13384934338031226048094025615, 3.43073818309742878040563403227, 4.22811156153174533081051037032, 5.11803002163668632979531790674, 5.69592348337304702412646216559, 6.17602596805243030125264376862, 6.82548206853928445309733859462, 7.72324807742982311876303502241
