| L(s) = 1 | − 0.0985·2-s − 1.99·4-s − 4.03·5-s − 4.56·7-s + 0.393·8-s + 0.397·10-s + 0.732·11-s + 2.33·13-s + 0.449·14-s + 3.94·16-s − 5.93·17-s + 5.44·19-s + 8.03·20-s − 0.0721·22-s − 1.94·23-s + 11.2·25-s − 0.230·26-s + 9.08·28-s + 29-s − 5.73·31-s − 1.17·32-s + 0.585·34-s + 18.4·35-s + 2.29·37-s − 0.535·38-s − 1.58·40-s − 8.10·41-s + ⋯ |
| L(s) = 1 | − 0.0696·2-s − 0.995·4-s − 1.80·5-s − 1.72·7-s + 0.138·8-s + 0.125·10-s + 0.220·11-s + 0.647·13-s + 0.120·14-s + 0.985·16-s − 1.44·17-s + 1.24·19-s + 1.79·20-s − 0.0153·22-s − 0.405·23-s + 2.25·25-s − 0.0451·26-s + 1.71·28-s + 0.185·29-s − 1.03·31-s − 0.207·32-s + 0.100·34-s + 3.11·35-s + 0.376·37-s − 0.0869·38-s − 0.250·40-s − 1.26·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 783 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 783 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4104202883\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4104202883\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + 0.0985T + 2T^{2} \) |
| 5 | \( 1 + 4.03T + 5T^{2} \) |
| 7 | \( 1 + 4.56T + 7T^{2} \) |
| 11 | \( 1 - 0.732T + 11T^{2} \) |
| 13 | \( 1 - 2.33T + 13T^{2} \) |
| 17 | \( 1 + 5.93T + 17T^{2} \) |
| 19 | \( 1 - 5.44T + 19T^{2} \) |
| 23 | \( 1 + 1.94T + 23T^{2} \) |
| 31 | \( 1 + 5.73T + 31T^{2} \) |
| 37 | \( 1 - 2.29T + 37T^{2} \) |
| 41 | \( 1 + 8.10T + 41T^{2} \) |
| 43 | \( 1 - 6.23T + 43T^{2} \) |
| 47 | \( 1 - 6.42T + 47T^{2} \) |
| 53 | \( 1 - 7.93T + 53T^{2} \) |
| 59 | \( 1 + 12.5T + 59T^{2} \) |
| 61 | \( 1 + 12.5T + 61T^{2} \) |
| 67 | \( 1 - 6.87T + 67T^{2} \) |
| 71 | \( 1 + 3.01T + 71T^{2} \) |
| 73 | \( 1 - 4.76T + 73T^{2} \) |
| 79 | \( 1 + 0.854T + 79T^{2} \) |
| 83 | \( 1 + 6.55T + 83T^{2} \) |
| 89 | \( 1 - 1.41T + 89T^{2} \) |
| 97 | \( 1 - 14.7T + 97T^{2} \) |
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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
−10.24445680035697911853610637523, −9.169325272836498815746098442920, −8.819158629679961826964514662574, −7.72260386921067616738245927215, −6.98741944355894741494541214662, −5.93912251418156909682914523686, −4.57512039786918757268031938974, −3.78031198472190204683467357830, −3.20375838943910458841034798807, −0.51631225622350115737203013108,
0.51631225622350115737203013108, 3.20375838943910458841034798807, 3.78031198472190204683467357830, 4.57512039786918757268031938974, 5.93912251418156909682914523686, 6.98741944355894741494541214662, 7.72260386921067616738245927215, 8.819158629679961826964514662574, 9.169325272836498815746098442920, 10.24445680035697911853610637523
