| L(s) = 1 | + 2.17·2-s + 2.70·4-s + 5-s − 0.539·7-s + 1.53·8-s + 2.17·10-s + 3.17·11-s + 4.87·13-s − 1.17·14-s − 2.07·16-s + 1.36·17-s − 19-s + 2.70·20-s + 6.87·22-s − 2.78·23-s + 25-s + 10.5·26-s − 1.46·28-s + 3.90·29-s − 2.44·31-s − 7.58·32-s + 2.97·34-s − 0.539·35-s − 4.14·37-s − 2.17·38-s + 1.53·40-s + 3.01·41-s + ⋯ |
| L(s) = 1 | + 1.53·2-s + 1.35·4-s + 0.447·5-s − 0.203·7-s + 0.544·8-s + 0.686·10-s + 0.955·11-s + 1.35·13-s − 0.312·14-s − 0.519·16-s + 0.332·17-s − 0.229·19-s + 0.605·20-s + 1.46·22-s − 0.581·23-s + 0.200·25-s + 2.07·26-s − 0.276·28-s + 0.725·29-s − 0.439·31-s − 1.34·32-s + 0.509·34-s − 0.0911·35-s − 0.680·37-s − 0.352·38-s + 0.243·40-s + 0.470·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.917268890\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.917268890\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - 2.17T + 2T^{2} \) |
| 7 | \( 1 + 0.539T + 7T^{2} \) |
| 11 | \( 1 - 3.17T + 11T^{2} \) |
| 13 | \( 1 - 4.87T + 13T^{2} \) |
| 17 | \( 1 - 1.36T + 17T^{2} \) |
| 23 | \( 1 + 2.78T + 23T^{2} \) |
| 29 | \( 1 - 3.90T + 29T^{2} \) |
| 31 | \( 1 + 2.44T + 31T^{2} \) |
| 37 | \( 1 + 4.14T + 37T^{2} \) |
| 41 | \( 1 - 3.01T + 41T^{2} \) |
| 43 | \( 1 - 5.95T + 43T^{2} \) |
| 47 | \( 1 + 4.04T + 47T^{2} \) |
| 53 | \( 1 - 6.63T + 53T^{2} \) |
| 59 | \( 1 + 12.4T + 59T^{2} \) |
| 61 | \( 1 + 9.31T + 61T^{2} \) |
| 67 | \( 1 + 7.75T + 67T^{2} \) |
| 71 | \( 1 - 2.18T + 71T^{2} \) |
| 73 | \( 1 + 7.60T + 73T^{2} \) |
| 79 | \( 1 + 15.0T + 79T^{2} \) |
| 83 | \( 1 - 2.78T + 83T^{2} \) |
| 89 | \( 1 - 0.829T + 89T^{2} \) |
| 97 | \( 1 + 5.37T + 97T^{2} \) |
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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
−10.42451204857941583339390214324, −9.305637220764868101314956437651, −8.571929981152227910014317445156, −7.23452881692479586839963732580, −6.15878017033444294061154119031, −5.99248299799533300705343977920, −4.71468501011512977055705515243, −3.86790672514301498744926717211, −3.04243558760489005769782222941, −1.61613616590321490615180463633,
1.61613616590321490615180463633, 3.04243558760489005769782222941, 3.86790672514301498744926717211, 4.71468501011512977055705515243, 5.99248299799533300705343977920, 6.15878017033444294061154119031, 7.23452881692479586839963732580, 8.571929981152227910014317445156, 9.305637220764868101314956437651, 10.42451204857941583339390214324
