| L(s) = 1 | − 0.596·2-s + 3.22·3-s − 1.64·4-s − 1.92·6-s − 0.0914·7-s + 2.17·8-s + 7.38·9-s + 0.938·11-s − 5.29·12-s − 6.40·13-s + 0.0545·14-s + 1.99·16-s + 5.47·17-s − 4.40·18-s − 4.62·19-s − 0.294·21-s − 0.559·22-s − 9.45·23-s + 7.00·24-s + 3.82·26-s + 14.1·27-s + 0.150·28-s − 0.666·29-s − 7.51·31-s − 5.53·32-s + 3.02·33-s − 3.26·34-s + ⋯ |
| L(s) = 1 | − 0.421·2-s + 1.86·3-s − 0.821·4-s − 0.784·6-s − 0.0345·7-s + 0.768·8-s + 2.46·9-s + 0.282·11-s − 1.52·12-s − 1.77·13-s + 0.0145·14-s + 0.497·16-s + 1.32·17-s − 1.03·18-s − 1.06·19-s − 0.0642·21-s − 0.119·22-s − 1.97·23-s + 1.43·24-s + 0.750·26-s + 2.71·27-s + 0.0284·28-s − 0.123·29-s − 1.34·31-s − 0.978·32-s + 0.526·33-s − 0.559·34-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 | \( 1 + 0.596T + 2T^{2} \) |
| 3 | \( 1 - 3.22T + 3T^{2} \) |
| 7 | \( 1 + 0.0914T + 7T^{2} \) |
| 11 | \( 1 - 0.938T + 11T^{2} \) |
| 13 | \( 1 + 6.40T + 13T^{2} \) |
| 17 | \( 1 - 5.47T + 17T^{2} \) |
| 19 | \( 1 + 4.62T + 19T^{2} \) |
| 23 | \( 1 + 9.45T + 23T^{2} \) |
| 29 | \( 1 + 0.666T + 29T^{2} \) |
| 31 | \( 1 + 7.51T + 31T^{2} \) |
| 37 | \( 1 + 5.97T + 37T^{2} \) |
| 41 | \( 1 + 1.82T + 41T^{2} \) |
| 43 | \( 1 + 2.41T + 43T^{2} \) |
| 47 | \( 1 + 1.50T + 47T^{2} \) |
| 53 | \( 1 - 3.52T + 53T^{2} \) |
| 59 | \( 1 + 14.3T + 59T^{2} \) |
| 61 | \( 1 + 6.52T + 61T^{2} \) |
| 67 | \( 1 - 15.3T + 67T^{2} \) |
| 71 | \( 1 - 6.49T + 71T^{2} \) |
| 73 | \( 1 - 2.73T + 73T^{2} \) |
| 79 | \( 1 + 8.80T + 79T^{2} \) |
| 83 | \( 1 - 0.402T + 83T^{2} \) |
| 89 | \( 1 - 17.9T + 89T^{2} \) |
| 97 | \( 1 + 17.4T + 97T^{2} \) |
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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
−7.81334465649811379791166878015, −7.55360895703518265604218782211, −6.61168498378648523191984969522, −5.36439687882692381985188737315, −4.59349206864955274691728592281, −3.85806147196046158112746984485, −3.30970917776606222313908554727, −2.20150822597081405799750352021, −1.63793348587671504969627480642, 0,
1.63793348587671504969627480642, 2.20150822597081405799750352021, 3.30970917776606222313908554727, 3.85806147196046158112746984485, 4.59349206864955274691728592281, 5.36439687882692381985188737315, 6.61168498378648523191984969522, 7.55360895703518265604218782211, 7.81334465649811379791166878015
