| L(s) = 1 | + 2-s + 2.90·3-s + 4-s + 0.347·5-s + 2.90·6-s + 1.00·7-s + 8-s + 5.45·9-s + 0.347·10-s − 5.46·11-s + 2.90·12-s + 6.36·13-s + 1.00·14-s + 1.00·15-s + 16-s + 0.522·17-s + 5.45·18-s + 5.00·19-s + 0.347·20-s + 2.93·21-s − 5.46·22-s + 1.44·23-s + 2.90·24-s − 4.87·25-s + 6.36·26-s + 7.13·27-s + 1.00·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.67·3-s + 0.5·4-s + 0.155·5-s + 1.18·6-s + 0.381·7-s + 0.353·8-s + 1.81·9-s + 0.109·10-s − 1.64·11-s + 0.839·12-s + 1.76·13-s + 0.269·14-s + 0.260·15-s + 0.250·16-s + 0.126·17-s + 1.28·18-s + 1.14·19-s + 0.0776·20-s + 0.640·21-s − 1.16·22-s + 0.300·23-s + 0.593·24-s − 0.975·25-s + 1.24·26-s + 1.37·27-s + 0.190·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2738 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2738 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.703362158\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.703362158\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 37 | \( 1 \) |
| good | 3 | \( 1 - 2.90T + 3T^{2} \) |
| 5 | \( 1 - 0.347T + 5T^{2} \) |
| 7 | \( 1 - 1.00T + 7T^{2} \) |
| 11 | \( 1 + 5.46T + 11T^{2} \) |
| 13 | \( 1 - 6.36T + 13T^{2} \) |
| 17 | \( 1 - 0.522T + 17T^{2} \) |
| 19 | \( 1 - 5.00T + 19T^{2} \) |
| 23 | \( 1 - 1.44T + 23T^{2} \) |
| 29 | \( 1 + 6.33T + 29T^{2} \) |
| 31 | \( 1 - 3.30T + 31T^{2} \) |
| 41 | \( 1 + 1.10T + 41T^{2} \) |
| 43 | \( 1 + 6.27T + 43T^{2} \) |
| 47 | \( 1 + 11.4T + 47T^{2} \) |
| 53 | \( 1 - 0.0922T + 53T^{2} \) |
| 59 | \( 1 - 2.99T + 59T^{2} \) |
| 61 | \( 1 - 3.81T + 61T^{2} \) |
| 67 | \( 1 + 1.16T + 67T^{2} \) |
| 71 | \( 1 - 0.312T + 71T^{2} \) |
| 73 | \( 1 - 13.2T + 73T^{2} \) |
| 79 | \( 1 - 9.30T + 79T^{2} \) |
| 83 | \( 1 - 9.46T + 83T^{2} \) |
| 89 | \( 1 + 15.7T + 89T^{2} \) |
| 97 | \( 1 + 0.929T + 97T^{2} \) |
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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
−8.581170707498073492617514302694, −8.000251360827081238893737200059, −7.64107223039384065064543677783, −6.57763898293353488827146025848, −5.56561265826313869469964378253, −4.86115276384965489602028912205, −3.65936046101162921097059661451, −3.31002078604316722507054260797, −2.33587664782211091778587878752, −1.48284063988042639018846749198,
1.48284063988042639018846749198, 2.33587664782211091778587878752, 3.31002078604316722507054260797, 3.65936046101162921097059661451, 4.86115276384965489602028912205, 5.56561265826313869469964378253, 6.57763898293353488827146025848, 7.64107223039384065064543677783, 8.000251360827081238893737200059, 8.581170707498073492617514302694
