| L(s) = 1 | + 0.167·2-s − 1.97·4-s − 3.80·5-s − 7-s − 0.665·8-s − 0.637·10-s − 11-s + 3.80·13-s − 0.167·14-s + 3.83·16-s − 0.334·17-s + 8.13·19-s + 7.50·20-s − 0.167·22-s + 1.66·23-s + 9.47·25-s + 0.637·26-s + 1.97·28-s − 0.195·29-s − 9.94·31-s + 1.97·32-s − 0.0560·34-s + 3.80·35-s − 4.47·37-s + 1.36·38-s + 2.53·40-s + 6.27·41-s + ⋯ |
| L(s) = 1 | + 0.118·2-s − 0.985·4-s − 1.70·5-s − 0.377·7-s − 0.235·8-s − 0.201·10-s − 0.301·11-s + 1.05·13-s − 0.0447·14-s + 0.958·16-s − 0.0812·17-s + 1.86·19-s + 1.67·20-s − 0.0357·22-s + 0.347·23-s + 1.89·25-s + 0.124·26-s + 0.372·28-s − 0.0363·29-s − 1.78·31-s + 0.348·32-s − 0.00961·34-s + 0.643·35-s − 0.735·37-s + 0.221·38-s + 0.400·40-s + 0.980·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7883082867\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7883082867\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 - 0.167T + 2T^{2} \) |
| 5 | \( 1 + 3.80T + 5T^{2} \) |
| 13 | \( 1 - 3.80T + 13T^{2} \) |
| 17 | \( 1 + 0.334T + 17T^{2} \) |
| 19 | \( 1 - 8.13T + 19T^{2} \) |
| 23 | \( 1 - 1.66T + 23T^{2} \) |
| 29 | \( 1 + 0.195T + 29T^{2} \) |
| 31 | \( 1 + 9.94T + 31T^{2} \) |
| 37 | \( 1 + 4.47T + 37T^{2} \) |
| 41 | \( 1 - 6.27T + 41T^{2} \) |
| 43 | \( 1 - 2.33T + 43T^{2} \) |
| 47 | \( 1 - 12.1T + 47T^{2} \) |
| 53 | \( 1 + 7.94T + 53T^{2} \) |
| 59 | \( 1 + 3.74T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 + 0.139T + 67T^{2} \) |
| 71 | \( 1 + 4.66T + 71T^{2} \) |
| 73 | \( 1 - 4.19T + 73T^{2} \) |
| 79 | \( 1 - 3.33T + 79T^{2} \) |
| 83 | \( 1 - 13.9T + 83T^{2} \) |
| 89 | \( 1 - 9.88T + 89T^{2} \) |
| 97 | \( 1 - 0.0560T + 97T^{2} \) |
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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
−10.61079187411746357050055282224, −9.350735673943346928605407243924, −8.791199579904368345814017220001, −7.79978326617433976009661469734, −7.27945009656910062558633431494, −5.80565782620660755873390959829, −4.84980053666902816021388289595, −3.76758633823650279172768131407, −3.32704169591219710646693010646, −0.74665830070033025775674089900,
0.74665830070033025775674089900, 3.32704169591219710646693010646, 3.76758633823650279172768131407, 4.84980053666902816021388289595, 5.80565782620660755873390959829, 7.27945009656910062558633431494, 7.79978326617433976009661469734, 8.791199579904368345814017220001, 9.350735673943346928605407243924, 10.61079187411746357050055282224
