| L(s) = 1 | − 2.12·2-s + 2.49·4-s + 2.47·5-s − 2.45·7-s − 1.05·8-s − 5.23·10-s + 4.01·11-s + 1.62·13-s + 5.20·14-s − 2.76·16-s − 4.89·17-s − 6.28·19-s + 6.16·20-s − 8.50·22-s + 3.83·23-s + 1.10·25-s − 3.44·26-s − 6.12·28-s + 0.357·29-s − 8.44·31-s + 7.96·32-s + 10.3·34-s − 6.06·35-s − 1.81·37-s + 13.3·38-s − 2.59·40-s − 10.7·41-s + ⋯ |
| L(s) = 1 | − 1.49·2-s + 1.24·4-s + 1.10·5-s − 0.927·7-s − 0.371·8-s − 1.65·10-s + 1.20·11-s + 0.450·13-s + 1.38·14-s − 0.690·16-s − 1.18·17-s − 1.44·19-s + 1.37·20-s − 1.81·22-s + 0.798·23-s + 0.220·25-s − 0.676·26-s − 1.15·28-s + 0.0663·29-s − 1.51·31-s + 1.40·32-s + 1.78·34-s − 1.02·35-s − 0.298·37-s + 2.16·38-s − 0.410·40-s − 1.67·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 239 | \( 1 - T \) |
| good | 2 | \( 1 + 2.12T + 2T^{2} \) |
| 5 | \( 1 - 2.47T + 5T^{2} \) |
| 7 | \( 1 + 2.45T + 7T^{2} \) |
| 11 | \( 1 - 4.01T + 11T^{2} \) |
| 13 | \( 1 - 1.62T + 13T^{2} \) |
| 17 | \( 1 + 4.89T + 17T^{2} \) |
| 19 | \( 1 + 6.28T + 19T^{2} \) |
| 23 | \( 1 - 3.83T + 23T^{2} \) |
| 29 | \( 1 - 0.357T + 29T^{2} \) |
| 31 | \( 1 + 8.44T + 31T^{2} \) |
| 37 | \( 1 + 1.81T + 37T^{2} \) |
| 41 | \( 1 + 10.7T + 41T^{2} \) |
| 43 | \( 1 - 9.09T + 43T^{2} \) |
| 47 | \( 1 + 11.0T + 47T^{2} \) |
| 53 | \( 1 + 5.78T + 53T^{2} \) |
| 59 | \( 1 - 1.43T + 59T^{2} \) |
| 61 | \( 1 - 2.21T + 61T^{2} \) |
| 67 | \( 1 + 14.2T + 67T^{2} \) |
| 71 | \( 1 - 4.12T + 71T^{2} \) |
| 73 | \( 1 - 2.78T + 73T^{2} \) |
| 79 | \( 1 - 7.84T + 79T^{2} \) |
| 83 | \( 1 - 9.91T + 83T^{2} \) |
| 89 | \( 1 + 0.978T + 89T^{2} \) |
| 97 | \( 1 + 16.8T + 97T^{2} \) |
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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
−8.971522014725962878691639390974, −8.282033800799448440018512941374, −6.97781655643342866845423055451, −6.60614557595836670900855575796, −5.99320254414350147469878186562, −4.65054950404136265284566193363, −3.52591331309711736655606884272, −2.20051754880824364210673340836, −1.51044915052966091444512786167, 0,
1.51044915052966091444512786167, 2.20051754880824364210673340836, 3.52591331309711736655606884272, 4.65054950404136265284566193363, 5.99320254414350147469878186562, 6.60614557595836670900855575796, 6.97781655643342866845423055451, 8.282033800799448440018512941374, 8.971522014725962878691639390974
