| L(s) = 1 | + 0.347·2-s − 1.87·3-s − 1.87·4-s − 0.652·6-s + 1.87·7-s − 1.34·8-s + 0.532·9-s + 0.879·11-s + 3.53·12-s + 1.18·13-s + 0.652·14-s + 3.29·16-s + 2.53·17-s + 0.184·18-s − 2.34·19-s − 3.53·21-s + 0.305·22-s − 0.347·23-s + 2.53·24-s + 0.411·26-s + 4.63·27-s − 3.53·28-s − 3.53·29-s − 5.36·31-s + 3.83·32-s − 1.65·33-s + 0.879·34-s + ⋯ |
| L(s) = 1 | + 0.245·2-s − 1.08·3-s − 0.939·4-s − 0.266·6-s + 0.710·7-s − 0.476·8-s + 0.177·9-s + 0.265·11-s + 1.01·12-s + 0.328·13-s + 0.174·14-s + 0.822·16-s + 0.614·17-s + 0.0435·18-s − 0.538·19-s − 0.770·21-s + 0.0651·22-s − 0.0724·23-s + 0.516·24-s + 0.0806·26-s + 0.892·27-s − 0.667·28-s − 0.655·29-s − 0.964·31-s + 0.678·32-s − 0.287·33-s + 0.150·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 47 | \( 1 - T \) |
| good | 2 | \( 1 - 0.347T + 2T^{2} \) |
| 3 | \( 1 + 1.87T + 3T^{2} \) |
| 7 | \( 1 - 1.87T + 7T^{2} \) |
| 11 | \( 1 - 0.879T + 11T^{2} \) |
| 13 | \( 1 - 1.18T + 13T^{2} \) |
| 17 | \( 1 - 2.53T + 17T^{2} \) |
| 19 | \( 1 + 2.34T + 19T^{2} \) |
| 23 | \( 1 + 0.347T + 23T^{2} \) |
| 29 | \( 1 + 3.53T + 29T^{2} \) |
| 31 | \( 1 + 5.36T + 31T^{2} \) |
| 37 | \( 1 + 3.24T + 37T^{2} \) |
| 41 | \( 1 + 10.4T + 41T^{2} \) |
| 43 | \( 1 - 1.18T + 43T^{2} \) |
| 53 | \( 1 + 1.77T + 53T^{2} \) |
| 59 | \( 1 + 7.59T + 59T^{2} \) |
| 61 | \( 1 + 2.24T + 61T^{2} \) |
| 67 | \( 1 + 8.35T + 67T^{2} \) |
| 71 | \( 1 + 3.73T + 71T^{2} \) |
| 73 | \( 1 - 4.63T + 73T^{2} \) |
| 79 | \( 1 + 4.16T + 79T^{2} \) |
| 83 | \( 1 - 1.46T + 83T^{2} \) |
| 89 | \( 1 - 11.7T + 89T^{2} \) |
| 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
−9.338593887658897971520808484419, −8.587359047434284380436858499314, −7.79584201622496082063675840417, −6.62806434078734510884396039831, −5.72194514005765186451973404987, −5.17204964553483171698317592774, −4.34808679304744708476102939646, −3.33653925844640676792214628731, −1.49010324496007804531367906657, 0,
1.49010324496007804531367906657, 3.33653925844640676792214628731, 4.34808679304744708476102939646, 5.17204964553483171698317592774, 5.72194514005765186451973404987, 6.62806434078734510884396039831, 7.79584201622496082063675840417, 8.587359047434284380436858499314, 9.338593887658897971520808484419
