| L(s) = 1 | + 0.859·2-s − 1.32·3-s − 1.26·4-s − 1.13·6-s − 0.529·7-s − 2.80·8-s − 1.24·9-s + 5.79·11-s + 1.67·12-s − 6.67·13-s − 0.455·14-s + 0.110·16-s + 4.32·17-s − 1.07·18-s + 3.78·19-s + 0.702·21-s + 4.98·22-s − 5.89·23-s + 3.71·24-s − 5.74·26-s + 5.62·27-s + 0.667·28-s − 1.38·29-s + 2.87·31-s + 5.70·32-s − 7.68·33-s + 3.71·34-s + ⋯ |
| L(s) = 1 | + 0.608·2-s − 0.764·3-s − 0.630·4-s − 0.465·6-s − 0.200·7-s − 0.991·8-s − 0.414·9-s + 1.74·11-s + 0.482·12-s − 1.85·13-s − 0.121·14-s + 0.0275·16-s + 1.04·17-s − 0.252·18-s + 0.869·19-s + 0.153·21-s + 1.06·22-s − 1.22·23-s + 0.758·24-s − 1.12·26-s + 1.08·27-s + 0.126·28-s − 0.256·29-s + 0.516·31-s + 1.00·32-s − 1.33·33-s + 0.637·34-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 241 | \( 1 + T \) |
| good | 2 | \( 1 - 0.859T + 2T^{2} \) |
| 3 | \( 1 + 1.32T + 3T^{2} \) |
| 7 | \( 1 + 0.529T + 7T^{2} \) |
| 11 | \( 1 - 5.79T + 11T^{2} \) |
| 13 | \( 1 + 6.67T + 13T^{2} \) |
| 17 | \( 1 - 4.32T + 17T^{2} \) |
| 19 | \( 1 - 3.78T + 19T^{2} \) |
| 23 | \( 1 + 5.89T + 23T^{2} \) |
| 29 | \( 1 + 1.38T + 29T^{2} \) |
| 31 | \( 1 - 2.87T + 31T^{2} \) |
| 37 | \( 1 + 4.89T + 37T^{2} \) |
| 41 | \( 1 - 6.31T + 41T^{2} \) |
| 43 | \( 1 - 3.37T + 43T^{2} \) |
| 47 | \( 1 + 10.5T + 47T^{2} \) |
| 53 | \( 1 - 7.63T + 53T^{2} \) |
| 59 | \( 1 - 15.1T + 59T^{2} \) |
| 61 | \( 1 - 1.41T + 61T^{2} \) |
| 67 | \( 1 + 1.62T + 67T^{2} \) |
| 71 | \( 1 - 0.160T + 71T^{2} \) |
| 73 | \( 1 - 1.85T + 73T^{2} \) |
| 79 | \( 1 + 4.62T + 79T^{2} \) |
| 83 | \( 1 - 10.8T + 83T^{2} \) |
| 89 | \( 1 - 6.27T + 89T^{2} \) |
| 97 | \( 1 + 16.7T + 97T^{2} \) |
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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
−7.63652581949227437951729830446, −6.77122949158455255691827851011, −6.16127407480340129241977978508, −5.42398007425651824893860388381, −4.99523391013489460388775307051, −4.10274129212502431710439348321, −3.47142460144189611876253600407, −2.52083836715215496170674502435, −1.09320602138542185399625294258, 0,
1.09320602138542185399625294258, 2.52083836715215496170674502435, 3.47142460144189611876253600407, 4.10274129212502431710439348321, 4.99523391013489460388775307051, 5.42398007425651824893860388381, 6.16127407480340129241977978508, 6.77122949158455255691827851011, 7.63652581949227437951729830446
