L(s) = 1 | + 1.73·2-s + 0.999·4-s − 1.73·5-s + 2·7-s − 1.73·8-s − 2.99·10-s − 3.46·11-s − 13-s + 3.46·14-s − 5·16-s + 5.19·17-s + 2·19-s − 1.73·20-s − 5.99·22-s + 3.46·23-s − 2.00·25-s − 1.73·26-s + 1.99·28-s + 1.73·29-s + 8·31-s − 5.19·32-s + 9·34-s − 3.46·35-s − 7·37-s + 3.46·38-s + 3.00·40-s − 6.92·41-s + ⋯ |
L(s) = 1 | + 1.22·2-s + 0.499·4-s − 0.774·5-s + 0.755·7-s − 0.612·8-s − 0.948·10-s − 1.04·11-s − 0.277·13-s + 0.925·14-s − 1.25·16-s + 1.26·17-s + 0.458·19-s − 0.387·20-s − 1.27·22-s + 0.722·23-s − 0.400·25-s − 0.339·26-s + 0.377·28-s + 0.321·29-s + 1.43·31-s − 0.918·32-s + 1.54·34-s − 0.585·35-s − 1.15·37-s + 0.561·38-s + 0.474·40-s − 1.08·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.435104167\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.435104167\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
good | 2 | \( 1 - 1.73T + 2T^{2} \) |
| 5 | \( 1 + 1.73T + 5T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 + 3.46T + 11T^{2} \) |
| 13 | \( 1 + T + 13T^{2} \) |
| 17 | \( 1 - 5.19T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 - 3.46T + 23T^{2} \) |
| 29 | \( 1 - 1.73T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + 7T + 37T^{2} \) |
| 41 | \( 1 + 6.92T + 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 - 6.92T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 13.8T + 59T^{2} \) |
| 61 | \( 1 + 7T + 61T^{2} \) |
| 67 | \( 1 + 10T + 67T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 + 7T + 73T^{2} \) |
| 79 | \( 1 - 2T + 79T^{2} \) |
| 83 | \( 1 + 13.8T + 83T^{2} \) |
| 89 | \( 1 + 5.19T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.29964332369256398717487679596, −13.39369631200027704092706736493, −12.22520072356073210718671997465, −11.60281525157102031033473031367, −10.20242410709674926896312591641, −8.462010452150115627851294225307, −7.36311860625225098919703816696, −5.56324876902780652194785160059, −4.60289706621031782825417838661, −3.12234009529020877746547298161,
3.12234009529020877746547298161, 4.60289706621031782825417838661, 5.56324876902780652194785160059, 7.36311860625225098919703816696, 8.462010452150115627851294225307, 10.20242410709674926896312591641, 11.60281525157102031033473031367, 12.22520072356073210718671997465, 13.39369631200027704092706736493, 14.29964332369256398717487679596