| L(s) = 1 | − 1.50·2-s − 0.655·3-s + 0.272·4-s − 5-s + 0.987·6-s − 1.30·7-s + 2.60·8-s − 2.57·9-s + 1.50·10-s − 11-s − 0.178·12-s + 2.03·13-s + 1.97·14-s + 0.655·15-s − 4.47·16-s + 0.00767·17-s + 3.87·18-s + 2.45·19-s − 0.272·20-s + 0.856·21-s + 1.50·22-s − 2.20·23-s − 1.70·24-s + 25-s − 3.06·26-s + 3.65·27-s − 0.356·28-s + ⋯ |
| L(s) = 1 | − 1.06·2-s − 0.378·3-s + 0.136·4-s − 0.447·5-s + 0.403·6-s − 0.494·7-s + 0.920·8-s − 0.856·9-s + 0.476·10-s − 0.301·11-s − 0.0515·12-s + 0.564·13-s + 0.526·14-s + 0.169·15-s − 1.11·16-s + 0.00186·17-s + 0.913·18-s + 0.562·19-s − 0.0609·20-s + 0.186·21-s + 0.321·22-s − 0.459·23-s − 0.348·24-s + 0.200·25-s − 0.601·26-s + 0.702·27-s − 0.0673·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 + T \) |
| good | 2 | \( 1 + 1.50T + 2T^{2} \) |
| 3 | \( 1 + 0.655T + 3T^{2} \) |
| 7 | \( 1 + 1.30T + 7T^{2} \) |
| 13 | \( 1 - 2.03T + 13T^{2} \) |
| 17 | \( 1 - 0.00767T + 17T^{2} \) |
| 19 | \( 1 - 2.45T + 19T^{2} \) |
| 23 | \( 1 + 2.20T + 23T^{2} \) |
| 29 | \( 1 + 5.12T + 29T^{2} \) |
| 31 | \( 1 + 3.16T + 31T^{2} \) |
| 37 | \( 1 - 7.40T + 37T^{2} \) |
| 41 | \( 1 - 5.71T + 41T^{2} \) |
| 43 | \( 1 + 2.78T + 43T^{2} \) |
| 47 | \( 1 - 7.47T + 47T^{2} \) |
| 53 | \( 1 + 5.37T + 53T^{2} \) |
| 59 | \( 1 - 9.19T + 59T^{2} \) |
| 61 | \( 1 - 3.65T + 61T^{2} \) |
| 67 | \( 1 - 8.26T + 67T^{2} \) |
| 71 | \( 1 + 1.32T + 71T^{2} \) |
| 79 | \( 1 - 9.22T + 79T^{2} \) |
| 83 | \( 1 + 17.6T + 83T^{2} \) |
| 89 | \( 1 - 7.40T + 89T^{2} \) |
| 97 | \( 1 - 7.13T + 97T^{2} \) |
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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
−8.104541880646113141204643255802, −7.62341232031926726013073450312, −6.77117744261663423808942319728, −5.90841458814293969671903400165, −5.22479192279729155626513432059, −4.20855545210589047644661229893, −3.39029081190946863973972598503, −2.29174287933099865889643576872, −0.959220052230229248093082183909, 0,
0.959220052230229248093082183909, 2.29174287933099865889643576872, 3.39029081190946863973972598503, 4.20855545210589047644661229893, 5.22479192279729155626513432059, 5.90841458814293969671903400165, 6.77117744261663423808942319728, 7.62341232031926726013073450312, 8.104541880646113141204643255802
