L(s) = 1 | + 2.58·2-s + 2.62·3-s + 4.69·4-s + 6.78·6-s + 4.29·7-s + 6.96·8-s + 3.87·9-s + 0.392·11-s + 12.3·12-s − 3.86·13-s + 11.1·14-s + 8.63·16-s − 2.28·17-s + 10.0·18-s − 6.51·19-s + 11.2·21-s + 1.01·22-s + 3.95·23-s + 18.2·24-s − 9.99·26-s + 2.30·27-s + 20.1·28-s − 2.20·29-s + 0.970·31-s + 8.40·32-s + 1.02·33-s − 5.90·34-s + ⋯ |
L(s) = 1 | + 1.82·2-s + 1.51·3-s + 2.34·4-s + 2.76·6-s + 1.62·7-s + 2.46·8-s + 1.29·9-s + 0.118·11-s + 3.55·12-s − 1.07·13-s + 2.97·14-s + 2.15·16-s − 0.553·17-s + 2.36·18-s − 1.49·19-s + 2.45·21-s + 0.216·22-s + 0.825·23-s + 3.72·24-s − 1.95·26-s + 0.442·27-s + 3.81·28-s − 0.410·29-s + 0.174·31-s + 1.48·32-s + 0.179·33-s − 1.01·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(12.61619167\) |
\(L(\frac12)\) |
\(\approx\) |
\(12.61619167\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 241 | \( 1 - T \) |
good | 2 | \( 1 - 2.58T + 2T^{2} \) |
| 3 | \( 1 - 2.62T + 3T^{2} \) |
| 7 | \( 1 - 4.29T + 7T^{2} \) |
| 11 | \( 1 - 0.392T + 11T^{2} \) |
| 13 | \( 1 + 3.86T + 13T^{2} \) |
| 17 | \( 1 + 2.28T + 17T^{2} \) |
| 19 | \( 1 + 6.51T + 19T^{2} \) |
| 23 | \( 1 - 3.95T + 23T^{2} \) |
| 29 | \( 1 + 2.20T + 29T^{2} \) |
| 31 | \( 1 - 0.970T + 31T^{2} \) |
| 37 | \( 1 + 3.07T + 37T^{2} \) |
| 41 | \( 1 - 8.48T + 41T^{2} \) |
| 43 | \( 1 + 4.56T + 43T^{2} \) |
| 47 | \( 1 + 10.9T + 47T^{2} \) |
| 53 | \( 1 + 0.970T + 53T^{2} \) |
| 59 | \( 1 - 2.47T + 59T^{2} \) |
| 61 | \( 1 + 8.72T + 61T^{2} \) |
| 67 | \( 1 - 14.8T + 67T^{2} \) |
| 71 | \( 1 - 7.87T + 71T^{2} \) |
| 73 | \( 1 + 5.53T + 73T^{2} \) |
| 79 | \( 1 - 1.20T + 79T^{2} \) |
| 83 | \( 1 + 10.3T + 83T^{2} \) |
| 89 | \( 1 + 15.3T + 89T^{2} \) |
| 97 | \( 1 - 6.20T + 97T^{2} \) |
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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
−8.005806086634725165860552373146, −7.25700015011211031091250006191, −6.72817835896664331416646044693, −5.67908614068000377208667428364, −4.77026210150926761433755704291, −4.53539825988385597405305614265, −3.74433902684181498814454640127, −2.86095004500016200685445710179, −2.17649099115214769400981464857, −1.71874695986199069007698875852,
1.71874695986199069007698875852, 2.17649099115214769400981464857, 2.86095004500016200685445710179, 3.74433902684181498814454640127, 4.53539825988385597405305614265, 4.77026210150926761433755704291, 5.67908614068000377208667428364, 6.72817835896664331416646044693, 7.25700015011211031091250006191, 8.005806086634725165860552373146