| L(s) = 1 | − 2-s − 2.37·3-s + 4-s − 2.30·5-s + 2.37·6-s + 0.129·7-s − 8-s + 2.62·9-s + 2.30·10-s + 5.58·11-s − 2.37·12-s + 2.97·13-s − 0.129·14-s + 5.47·15-s + 16-s − 0.801·17-s − 2.62·18-s + 19-s − 2.30·20-s − 0.306·21-s − 5.58·22-s − 0.622·23-s + 2.37·24-s + 0.325·25-s − 2.97·26-s + 0.885·27-s + 0.129·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.36·3-s + 0.5·4-s − 1.03·5-s + 0.968·6-s + 0.0488·7-s − 0.353·8-s + 0.875·9-s + 0.729·10-s + 1.68·11-s − 0.684·12-s + 0.826·13-s − 0.0345·14-s + 1.41·15-s + 0.250·16-s − 0.194·17-s − 0.619·18-s + 0.229·19-s − 0.515·20-s − 0.0668·21-s − 1.19·22-s − 0.129·23-s + 0.484·24-s + 0.0650·25-s − 0.584·26-s + 0.170·27-s + 0.0244·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 211 | \( 1 - T \) |
| good | 3 | \( 1 + 2.37T + 3T^{2} \) |
| 5 | \( 1 + 2.30T + 5T^{2} \) |
| 7 | \( 1 - 0.129T + 7T^{2} \) |
| 11 | \( 1 - 5.58T + 11T^{2} \) |
| 13 | \( 1 - 2.97T + 13T^{2} \) |
| 17 | \( 1 + 0.801T + 17T^{2} \) |
| 23 | \( 1 + 0.622T + 23T^{2} \) |
| 29 | \( 1 + 2.21T + 29T^{2} \) |
| 31 | \( 1 - 1.01T + 31T^{2} \) |
| 37 | \( 1 + 5.12T + 37T^{2} \) |
| 41 | \( 1 + 10.1T + 41T^{2} \) |
| 43 | \( 1 + 2.32T + 43T^{2} \) |
| 47 | \( 1 - 11.3T + 47T^{2} \) |
| 53 | \( 1 + 2.55T + 53T^{2} \) |
| 59 | \( 1 - 8.14T + 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 - 3.73T + 67T^{2} \) |
| 71 | \( 1 - 4.94T + 71T^{2} \) |
| 73 | \( 1 - 0.583T + 73T^{2} \) |
| 79 | \( 1 + 2.00T + 79T^{2} \) |
| 83 | \( 1 + 17.2T + 83T^{2} \) |
| 89 | \( 1 - 13.0T + 89T^{2} \) |
| 97 | \( 1 - 1.54T + 97T^{2} \) |
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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
−7.33886823945110142501755712608, −6.77607168721685287076162195585, −6.27344522902420109332427709377, −5.58760062054352328290819385947, −4.67321664675337202212160522331, −3.91100417933349003986233012746, −3.32316398021758917298099771572, −1.74959393386387312156331126423, −0.970000821772459409234022696118, 0,
0.970000821772459409234022696118, 1.74959393386387312156331126423, 3.32316398021758917298099771572, 3.91100417933349003986233012746, 4.67321664675337202212160522331, 5.58760062054352328290819385947, 6.27344522902420109332427709377, 6.77607168721685287076162195585, 7.33886823945110142501755712608
