| L(s) = 1 | + 2-s + 4-s + 2·5-s + 8-s + 2·10-s + 2·11-s − 4·13-s + 16-s − 19-s + 2·20-s + 2·22-s + 8·23-s − 25-s − 4·26-s − 2·29-s − 2·31-s + 32-s − 8·37-s − 38-s + 2·40-s − 2·41-s + 4·43-s + 2·44-s + 8·46-s − 4·47-s − 7·49-s − 50-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s + 0.353·8-s + 0.632·10-s + 0.603·11-s − 1.10·13-s + 1/4·16-s − 0.229·19-s + 0.447·20-s + 0.426·22-s + 1.66·23-s − 1/5·25-s − 0.784·26-s − 0.371·29-s − 0.359·31-s + 0.176·32-s − 1.31·37-s − 0.162·38-s + 0.316·40-s − 0.312·41-s + 0.609·43-s + 0.301·44-s + 1.17·46-s − 0.583·47-s − 49-s − 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.237563715\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.237563715\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−11.65314297553202694819366730155, −10.67137801203704435709118691628, −9.726493160639635780030466105686, −8.902474420814486458148417198970, −7.43472461846693450494480621223, −6.59001220829617312734896517440, −5.53295861295123136716745752833, −4.64180517544954192308318972299, −3.17827142125620977279897528792, −1.85086593928300837816640591535,
1.85086593928300837816640591535, 3.17827142125620977279897528792, 4.64180517544954192308318972299, 5.53295861295123136716745752833, 6.59001220829617312734896517440, 7.43472461846693450494480621223, 8.902474420814486458148417198970, 9.726493160639635780030466105686, 10.67137801203704435709118691628, 11.65314297553202694819366730155
