| L(s) = 1 | + 2.56·2-s + 4.56·4-s + 5-s + 0.438·7-s + 6.56·8-s + 2.56·10-s − 1.56·11-s + 1.12·14-s + 7.68·16-s − 1.56·17-s + 5.12·19-s + 4.56·20-s − 4·22-s − 2.43·23-s + 25-s + 2·28-s + 7.12·29-s − 6·31-s + 6.56·32-s − 4·34-s + 0.438·35-s + 10.6·37-s + 13.1·38-s + 6.56·40-s + 3.56·41-s + 3.12·43-s − 7.12·44-s + ⋯ |
| L(s) = 1 | + 1.81·2-s + 2.28·4-s + 0.447·5-s + 0.165·7-s + 2.31·8-s + 0.810·10-s − 0.470·11-s + 0.300·14-s + 1.92·16-s − 0.378·17-s + 1.17·19-s + 1.01·20-s − 0.852·22-s − 0.508·23-s + 0.200·25-s + 0.377·28-s + 1.32·29-s − 1.07·31-s + 1.15·32-s − 0.685·34-s + 0.0741·35-s + 1.75·37-s + 2.12·38-s + 1.03·40-s + 0.556·41-s + 0.476·43-s − 1.07·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.826678273\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.826678273\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 2.56T + 2T^{2} \) |
| 7 | \( 1 - 0.438T + 7T^{2} \) |
| 11 | \( 1 + 1.56T + 11T^{2} \) |
| 17 | \( 1 + 1.56T + 17T^{2} \) |
| 19 | \( 1 - 5.12T + 19T^{2} \) |
| 23 | \( 1 + 2.43T + 23T^{2} \) |
| 29 | \( 1 - 7.12T + 29T^{2} \) |
| 31 | \( 1 + 6T + 31T^{2} \) |
| 37 | \( 1 - 10.6T + 37T^{2} \) |
| 41 | \( 1 - 3.56T + 41T^{2} \) |
| 43 | \( 1 - 3.12T + 43T^{2} \) |
| 47 | \( 1 - 11.1T + 47T^{2} \) |
| 53 | \( 1 - 4.68T + 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 + 6.68T + 61T^{2} \) |
| 67 | \( 1 - 11.3T + 67T^{2} \) |
| 71 | \( 1 - 10.4T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 - 4.68T + 79T^{2} \) |
| 83 | \( 1 + 16.4T + 83T^{2} \) |
| 89 | \( 1 - 10.6T + 89T^{2} \) |
| 97 | \( 1 + 16.9T + 97T^{2} \) |
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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
−7.59321575760873237931065931379, −6.92349524933739744636195677300, −6.25706943984766430329313840289, −5.55027474540864750263646498083, −5.16880896885521696592944676393, −4.31631928167035812072688168371, −3.73457767706771014334779600878, −2.70738654360652548631348999353, −2.34208269898139626415131954208, −1.12077187945125914482194410817,
1.12077187945125914482194410817, 2.34208269898139626415131954208, 2.70738654360652548631348999353, 3.73457767706771014334779600878, 4.31631928167035812072688168371, 5.16880896885521696592944676393, 5.55027474540864750263646498083, 6.25706943984766430329313840289, 6.92349524933739744636195677300, 7.59321575760873237931065931379
