| L(s) = 1 | − 1.30·2-s − 0.302·4-s − 5-s + 4.60·7-s + 3·8-s + 1.30·10-s + 2.60·11-s − 0.605·13-s − 6·14-s − 3.30·16-s + 5.60·17-s − 3.60·19-s + 0.302·20-s − 3.39·22-s + 3·23-s + 25-s + 0.788·26-s − 1.39·28-s − 8.60·29-s + 1.60·31-s − 1.69·32-s − 7.30·34-s − 4.60·35-s + 2·37-s + 4.69·38-s − 3·40-s − 2.60·41-s + ⋯ |
| L(s) = 1 | − 0.921·2-s − 0.151·4-s − 0.447·5-s + 1.74·7-s + 1.06·8-s + 0.411·10-s + 0.785·11-s − 0.167·13-s − 1.60·14-s − 0.825·16-s + 1.35·17-s − 0.827·19-s + 0.0677·20-s − 0.723·22-s + 0.625·23-s + 0.200·25-s + 0.154·26-s − 0.263·28-s − 1.59·29-s + 0.288·31-s − 0.300·32-s − 1.25·34-s − 0.778·35-s + 0.328·37-s + 0.761·38-s − 0.474·40-s − 0.406·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7203246596\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7203246596\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| good | 2 | \( 1 + 1.30T + 2T^{2} \) |
| 7 | \( 1 - 4.60T + 7T^{2} \) |
| 11 | \( 1 - 2.60T + 11T^{2} \) |
| 13 | \( 1 + 0.605T + 13T^{2} \) |
| 17 | \( 1 - 5.60T + 17T^{2} \) |
| 19 | \( 1 + 3.60T + 19T^{2} \) |
| 23 | \( 1 - 3T + 23T^{2} \) |
| 29 | \( 1 + 8.60T + 29T^{2} \) |
| 31 | \( 1 - 1.60T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + 2.60T + 41T^{2} \) |
| 43 | \( 1 + 6.60T + 43T^{2} \) |
| 47 | \( 1 + 5.21T + 47T^{2} \) |
| 53 | \( 1 - 5.60T + 53T^{2} \) |
| 59 | \( 1 - 8.60T + 59T^{2} \) |
| 61 | \( 1 - 10.2T + 61T^{2} \) |
| 67 | \( 1 + 15.2T + 67T^{2} \) |
| 71 | \( 1 + 14.6T + 71T^{2} \) |
| 73 | \( 1 - 5.39T + 73T^{2} \) |
| 79 | \( 1 + 4.39T + 79T^{2} \) |
| 83 | \( 1 + 3T + 83T^{2} \) |
| 89 | \( 1 + 7.81T + 89T^{2} \) |
| 97 | \( 1 - 8T + 97T^{2} \) |
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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
−13.26651845725768254820666044922, −11.86950036690744818610321139579, −11.13649202637338755926750469179, −10.07946073498264653660591120770, −8.851300688689427455059761656759, −8.108019010147471591376132854846, −7.25188465063029580597412460768, −5.24881298504736273695468213845, −4.10459038170170635162919227844, −1.48047827314022114275690514485,
1.48047827314022114275690514485, 4.10459038170170635162919227844, 5.24881298504736273695468213845, 7.25188465063029580597412460768, 8.108019010147471591376132854846, 8.851300688689427455059761656759, 10.07946073498264653660591120770, 11.13649202637338755926750469179, 11.86950036690744818610321139579, 13.26651845725768254820666044922
