| L(s) = 1 | − 1.29·2-s − 3·3-s − 6.31·4-s + 3.89·6-s − 34.4·7-s + 18.5·8-s + 9·9-s + 7.37·11-s + 18.9·12-s + 13·13-s + 44.7·14-s + 26.3·16-s − 67.1·17-s − 11.6·18-s − 63.6·19-s + 103.·21-s − 9.58·22-s − 165.·23-s − 55.7·24-s − 16.8·26-s − 27·27-s + 217.·28-s − 212.·29-s − 230.·31-s − 182.·32-s − 22.1·33-s + 87.2·34-s + ⋯ |
| L(s) = 1 | − 0.459·2-s − 0.577·3-s − 0.788·4-s + 0.265·6-s − 1.86·7-s + 0.821·8-s + 0.333·9-s + 0.202·11-s + 0.455·12-s + 0.277·13-s + 0.854·14-s + 0.411·16-s − 0.958·17-s − 0.153·18-s − 0.768·19-s + 1.07·21-s − 0.0928·22-s − 1.49·23-s − 0.474·24-s − 0.127·26-s − 0.192·27-s + 1.46·28-s − 1.35·29-s − 1.33·31-s − 1.01·32-s − 0.116·33-s + 0.440·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.01984909760\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01984909760\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - 13T \) |
| good | 2 | \( 1 + 1.29T + 8T^{2} \) |
| 7 | \( 1 + 34.4T + 343T^{2} \) |
| 11 | \( 1 - 7.37T + 1.33e3T^{2} \) |
| 17 | \( 1 + 67.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 63.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + 165.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 212.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 230.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 316.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 76.4T + 6.89e4T^{2} \) |
| 43 | \( 1 + 267.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 47.5T + 1.03e5T^{2} \) |
| 53 | \( 1 + 426.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 818.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 127.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 538.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 221.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 145.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.27e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.01e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.44e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.00e3T + 9.12e5T^{2} \) |
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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
−9.444627901801938689037979310297, −9.152389958400696500311464557102, −8.053824260255676824571561486464, −6.95796536740837933836889992265, −6.27756095240972916066453616083, −5.42206600883774638964994782960, −4.14465229771679204964048719149, −3.52559272524580193682709146795, −1.85014645824918393484220647529, −0.080727321434615411005369414873,
0.080727321434615411005369414873, 1.85014645824918393484220647529, 3.52559272524580193682709146795, 4.14465229771679204964048719149, 5.42206600883774638964994782960, 6.27756095240972916066453616083, 6.95796536740837933836889992265, 8.053824260255676824571561486464, 9.152389958400696500311464557102, 9.444627901801938689037979310297
