| L(s) = 1 | − 2.41·2-s + 3.85·4-s + 3.18·7-s − 4.49·8-s − 4.15·11-s − 2.07·13-s − 7.71·14-s + 3.15·16-s + 5.79·17-s − 19-s + 10.0·22-s − 2.60·23-s + 5.01·26-s + 12.2·28-s − 6·29-s + 2.59·31-s + 1.34·32-s − 14.0·34-s − 4.30·37-s + 2.41·38-s + 0.599·41-s − 3.18·43-s − 16.0·44-s + 6.31·46-s + 11.7·47-s + 3.15·49-s − 7.98·52-s + ⋯ |
| L(s) = 1 | − 1.71·2-s + 1.92·4-s + 1.20·7-s − 1.58·8-s − 1.25·11-s − 0.574·13-s − 2.06·14-s + 0.788·16-s + 1.40·17-s − 0.229·19-s + 2.14·22-s − 0.543·23-s + 0.982·26-s + 2.32·28-s − 1.11·29-s + 0.466·31-s + 0.237·32-s − 2.40·34-s − 0.707·37-s + 0.392·38-s + 0.0935·41-s − 0.485·43-s − 2.41·44-s + 0.930·46-s + 1.70·47-s + 0.450·49-s − 1.10·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + 2.41T + 2T^{2} \) |
| 7 | \( 1 - 3.18T + 7T^{2} \) |
| 11 | \( 1 + 4.15T + 11T^{2} \) |
| 13 | \( 1 + 2.07T + 13T^{2} \) |
| 17 | \( 1 - 5.79T + 17T^{2} \) |
| 23 | \( 1 + 2.60T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 2.59T + 31T^{2} \) |
| 37 | \( 1 + 4.30T + 37T^{2} \) |
| 41 | \( 1 - 0.599T + 41T^{2} \) |
| 43 | \( 1 + 3.18T + 43T^{2} \) |
| 47 | \( 1 - 11.7T + 47T^{2} \) |
| 53 | \( 1 - 11.7T + 53T^{2} \) |
| 59 | \( 1 - 1.71T + 59T^{2} \) |
| 61 | \( 1 + 8.75T + 61T^{2} \) |
| 67 | \( 1 + 4.76T + 67T^{2} \) |
| 71 | \( 1 + 13.7T + 71T^{2} \) |
| 73 | \( 1 - 2.72T + 73T^{2} \) |
| 79 | \( 1 + 1.40T + 79T^{2} \) |
| 83 | \( 1 + 7.07T + 83T^{2} \) |
| 89 | \( 1 + 16.5T + 89T^{2} \) |
| 97 | \( 1 + 2.07T + 97T^{2} \) |
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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
−8.095153400970106343654582775647, −7.52058645647932586581091987705, −7.15211589177088900137286099249, −5.82463360428793627438860736952, −5.28962637364928448991156543287, −4.26080223924823850499479973217, −2.90226219681943277090419781772, −2.09665053626398941636759722964, −1.25336720337519930728368557478, 0,
1.25336720337519930728368557478, 2.09665053626398941636759722964, 2.90226219681943277090419781772, 4.26080223924823850499479973217, 5.28962637364928448991156543287, 5.82463360428793627438860736952, 7.15211589177088900137286099249, 7.52058645647932586581091987705, 8.095153400970106343654582775647
