| L(s) = 1 | + 2.49·2-s + 4.20·4-s − 1.49·5-s + 1.42·7-s + 5.49·8-s − 3.71·10-s + 1.42·11-s + 4·13-s + 3.55·14-s + 5.26·16-s − 1.42·17-s + 3.28·19-s − 6.26·20-s + 3.55·22-s − 8.40·23-s − 2.77·25-s + 9.96·26-s + 5.99·28-s + 3.77·29-s + 2.98·31-s + 2.14·32-s − 3.55·34-s − 2.12·35-s − 3.12·37-s + 8.18·38-s − 8.18·40-s − 5.42·41-s + ⋯ |
| L(s) = 1 | + 1.76·2-s + 2.10·4-s − 0.666·5-s + 0.539·7-s + 1.94·8-s − 1.17·10-s + 0.430·11-s + 1.10·13-s + 0.950·14-s + 1.31·16-s − 0.346·17-s + 0.753·19-s − 1.40·20-s + 0.757·22-s − 1.75·23-s − 0.555·25-s + 1.95·26-s + 1.13·28-s + 0.701·29-s + 0.535·31-s + 0.378·32-s − 0.609·34-s − 0.359·35-s − 0.513·37-s + 1.32·38-s − 1.29·40-s − 0.847·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.944165630\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.944165630\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 71 | \( 1 + T \) |
| good | 2 | \( 1 - 2.49T + 2T^{2} \) |
| 5 | \( 1 + 1.49T + 5T^{2} \) |
| 7 | \( 1 - 1.42T + 7T^{2} \) |
| 11 | \( 1 - 1.42T + 11T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 + 1.42T + 17T^{2} \) |
| 19 | \( 1 - 3.28T + 19T^{2} \) |
| 23 | \( 1 + 8.40T + 23T^{2} \) |
| 29 | \( 1 - 3.77T + 29T^{2} \) |
| 31 | \( 1 - 2.98T + 31T^{2} \) |
| 37 | \( 1 + 3.12T + 37T^{2} \) |
| 41 | \( 1 + 5.42T + 41T^{2} \) |
| 43 | \( 1 + 3.93T + 43T^{2} \) |
| 47 | \( 1 + 2.40T + 47T^{2} \) |
| 53 | \( 1 + 4.98T + 53T^{2} \) |
| 59 | \( 1 - 6.57T + 59T^{2} \) |
| 61 | \( 1 - 6.12T + 61T^{2} \) |
| 67 | \( 1 + 13.9T + 67T^{2} \) |
| 73 | \( 1 + 1.49T + 73T^{2} \) |
| 79 | \( 1 - 14.0T + 79T^{2} \) |
| 83 | \( 1 + 7.28T + 83T^{2} \) |
| 89 | \( 1 - 5.04T + 89T^{2} \) |
| 97 | \( 1 - 5.55T + 97T^{2} \) |
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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
−11.07054654150751456702814153608, −10.02528716902680486909776887111, −8.543761440865789507487421945804, −7.75033150869000990107700095254, −6.64615308753169699810206769219, −5.92934253808867245680933229978, −4.86660811620888514279607579787, −4.02630161496223896925951913229, −3.30942772524896953204734872967, −1.79832937123606872317699577714,
1.79832937123606872317699577714, 3.30942772524896953204734872967, 4.02630161496223896925951913229, 4.86660811620888514279607579787, 5.92934253808867245680933229978, 6.64615308753169699810206769219, 7.75033150869000990107700095254, 8.543761440865789507487421945804, 10.02528716902680486909776887111, 11.07054654150751456702814153608
