| L(s) = 1 | − 0.787·2-s − 1.38·4-s + 3.87·5-s − 4.77·7-s + 2.66·8-s − 3.04·10-s − 2.02·11-s − 1.04·13-s + 3.76·14-s + 0.666·16-s + 4.63·19-s − 5.34·20-s + 1.59·22-s + 5.29·23-s + 10.0·25-s + 0.820·26-s + 6.59·28-s − 7.94·29-s − 5.04·31-s − 5.84·32-s − 18.5·35-s + 1.43·37-s − 3.64·38-s + 10.3·40-s − 7.68·41-s + 1.23·43-s + 2.79·44-s + ⋯ |
| L(s) = 1 | − 0.556·2-s − 0.690·4-s + 1.73·5-s − 1.80·7-s + 0.940·8-s − 0.964·10-s − 0.609·11-s − 0.288·13-s + 1.00·14-s + 0.166·16-s + 1.06·19-s − 1.19·20-s + 0.339·22-s + 1.10·23-s + 2.00·25-s + 0.160·26-s + 1.24·28-s − 1.47·29-s − 0.906·31-s − 1.03·32-s − 3.12·35-s + 0.235·37-s − 0.591·38-s + 1.63·40-s − 1.20·41-s + 0.188·43-s + 0.420·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7803 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7803 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + 0.787T + 2T^{2} \) |
| 5 | \( 1 - 3.87T + 5T^{2} \) |
| 7 | \( 1 + 4.77T + 7T^{2} \) |
| 11 | \( 1 + 2.02T + 11T^{2} \) |
| 13 | \( 1 + 1.04T + 13T^{2} \) |
| 19 | \( 1 - 4.63T + 19T^{2} \) |
| 23 | \( 1 - 5.29T + 23T^{2} \) |
| 29 | \( 1 + 7.94T + 29T^{2} \) |
| 31 | \( 1 + 5.04T + 31T^{2} \) |
| 37 | \( 1 - 1.43T + 37T^{2} \) |
| 41 | \( 1 + 7.68T + 41T^{2} \) |
| 43 | \( 1 - 1.23T + 43T^{2} \) |
| 47 | \( 1 - 1.60T + 47T^{2} \) |
| 53 | \( 1 - 5.76T + 53T^{2} \) |
| 59 | \( 1 - 2.47T + 59T^{2} \) |
| 61 | \( 1 + 8.81T + 61T^{2} \) |
| 67 | \( 1 - 9.68T + 67T^{2} \) |
| 71 | \( 1 - 2.83T + 71T^{2} \) |
| 73 | \( 1 + 5.32T + 73T^{2} \) |
| 79 | \( 1 - 10.1T + 79T^{2} \) |
| 83 | \( 1 + 10.2T + 83T^{2} \) |
| 89 | \( 1 + 1.39T + 89T^{2} \) |
| 97 | \( 1 - 0.0774T + 97T^{2} \) |
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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
−7.29433842854703208929177844166, −6.98826833615201090642833374685, −6.00117488492092168507301717344, −5.49652649554831375842015370331, −4.98890419080487956980218027582, −3.72131594623945799528777661377, −3.03316542833811675951784420045, −2.18855757387694615270416371211, −1.13420469858731210467371063190, 0,
1.13420469858731210467371063190, 2.18855757387694615270416371211, 3.03316542833811675951784420045, 3.72131594623945799528777661377, 4.98890419080487956980218027582, 5.49652649554831375842015370331, 6.00117488492092168507301717344, 6.98826833615201090642833374685, 7.29433842854703208929177844166
