| L(s) = 1 | − 2.30·2-s − 1.94·3-s + 3.32·4-s + 4.48·6-s − 0.292·7-s − 3.07·8-s + 0.779·9-s + 5.73·11-s − 6.47·12-s − 6.40·13-s + 0.676·14-s + 0.428·16-s + 1.19·17-s − 1.79·18-s − 6.61·19-s + 0.569·21-s − 13.2·22-s − 4.21·23-s + 5.96·24-s + 14.7·26-s + 4.31·27-s − 0.975·28-s + 7.24·29-s + 5.26·31-s + 5.15·32-s − 11.1·33-s − 2.76·34-s + ⋯ |
| L(s) = 1 | − 1.63·2-s − 1.12·3-s + 1.66·4-s + 1.83·6-s − 0.110·7-s − 1.08·8-s + 0.259·9-s + 1.73·11-s − 1.86·12-s − 1.77·13-s + 0.180·14-s + 0.107·16-s + 0.290·17-s − 0.424·18-s − 1.51·19-s + 0.124·21-s − 2.82·22-s − 0.878·23-s + 1.21·24-s + 2.90·26-s + 0.830·27-s − 0.184·28-s + 1.34·29-s + 0.945·31-s + 0.910·32-s − 1.94·33-s − 0.474·34-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 | \( 1 + 2.30T + 2T^{2} \) |
| 3 | \( 1 + 1.94T + 3T^{2} \) |
| 7 | \( 1 + 0.292T + 7T^{2} \) |
| 11 | \( 1 - 5.73T + 11T^{2} \) |
| 13 | \( 1 + 6.40T + 13T^{2} \) |
| 17 | \( 1 - 1.19T + 17T^{2} \) |
| 19 | \( 1 + 6.61T + 19T^{2} \) |
| 23 | \( 1 + 4.21T + 23T^{2} \) |
| 29 | \( 1 - 7.24T + 29T^{2} \) |
| 31 | \( 1 - 5.26T + 31T^{2} \) |
| 37 | \( 1 - 5.15T + 37T^{2} \) |
| 41 | \( 1 + 4.48T + 41T^{2} \) |
| 43 | \( 1 - 1.68T + 43T^{2} \) |
| 47 | \( 1 - 6.59T + 47T^{2} \) |
| 53 | \( 1 - 0.965T + 53T^{2} \) |
| 59 | \( 1 - 3.89T + 59T^{2} \) |
| 61 | \( 1 + 6.97T + 61T^{2} \) |
| 67 | \( 1 + 12.8T + 67T^{2} \) |
| 71 | \( 1 + 14.7T + 71T^{2} \) |
| 73 | \( 1 + 6.70T + 73T^{2} \) |
| 79 | \( 1 - 11.4T + 79T^{2} \) |
| 83 | \( 1 - 5.98T + 83T^{2} \) |
| 89 | \( 1 - 6.24T + 89T^{2} \) |
| 97 | \( 1 - 4.42T + 97T^{2} \) |
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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
−7.76153593761624567545732269994, −7.05326588823010430578560291587, −6.33840379681883022360326699692, −6.14046799925109518527485822082, −4.76738804378659402644117725696, −4.31391851875837219279220058253, −2.82962576409847561394312049712, −1.91610475341219528591675871065, −0.903779332179725701059488084899, 0,
0.903779332179725701059488084899, 1.91610475341219528591675871065, 2.82962576409847561394312049712, 4.31391851875837219279220058253, 4.76738804378659402644117725696, 6.14046799925109518527485822082, 6.33840379681883022360326699692, 7.05326588823010430578560291587, 7.76153593761624567545732269994
