| L(s) = 1 | − 0.445·2-s − 1.80·4-s + 5-s − 4·7-s + 1.69·8-s − 0.445·10-s + 4.71·11-s + 1.78·14-s + 2.85·16-s − 1.95·17-s − 7.34·19-s − 1.80·20-s − 2.09·22-s + 2.02·23-s + 25-s + 7.20·28-s − 6.98·29-s + 7.82·31-s − 4.65·32-s + 0.868·34-s − 4·35-s + 10.3·37-s + 3.26·38-s + 1.69·40-s − 10.8·41-s + 4.27·43-s − 8.49·44-s + ⋯ |
| L(s) = 1 | − 0.314·2-s − 0.900·4-s + 0.447·5-s − 1.51·7-s + 0.598·8-s − 0.140·10-s + 1.42·11-s + 0.475·14-s + 0.712·16-s − 0.473·17-s − 1.68·19-s − 0.402·20-s − 0.447·22-s + 0.421·23-s + 0.200·25-s + 1.36·28-s − 1.29·29-s + 1.40·31-s − 0.822·32-s + 0.148·34-s − 0.676·35-s + 1.70·37-s + 0.530·38-s + 0.267·40-s − 1.68·41-s + 0.651·43-s − 1.28·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 0.445T + 2T^{2} \) |
| 7 | \( 1 + 4T + 7T^{2} \) |
| 11 | \( 1 - 4.71T + 11T^{2} \) |
| 17 | \( 1 + 1.95T + 17T^{2} \) |
| 19 | \( 1 + 7.34T + 19T^{2} \) |
| 23 | \( 1 - 2.02T + 23T^{2} \) |
| 29 | \( 1 + 6.98T + 29T^{2} \) |
| 31 | \( 1 - 7.82T + 31T^{2} \) |
| 37 | \( 1 - 10.3T + 37T^{2} \) |
| 41 | \( 1 + 10.8T + 41T^{2} \) |
| 43 | \( 1 - 4.27T + 43T^{2} \) |
| 47 | \( 1 + 1.07T + 47T^{2} \) |
| 53 | \( 1 - 8.23T + 53T^{2} \) |
| 59 | \( 1 - 13.7T + 59T^{2} \) |
| 61 | \( 1 + 2.76T + 61T^{2} \) |
| 67 | \( 1 + 11.5T + 67T^{2} \) |
| 71 | \( 1 + 7.20T + 71T^{2} \) |
| 73 | \( 1 - 1.60T + 73T^{2} \) |
| 79 | \( 1 + 8.47T + 79T^{2} \) |
| 83 | \( 1 - 1.87T + 83T^{2} \) |
| 89 | \( 1 + 3.32T + 89T^{2} \) |
| 97 | \( 1 - 9.60T + 97T^{2} \) |
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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
−7.49403317343819764190027743942, −6.63963476205521154352416039535, −6.31776468449709642497610278420, −5.57897628140360803884546521714, −4.43642775208533803047423978477, −4.05211855016842051337082823228, −3.21902491428633574642340557123, −2.19778253531805911406355116416, −1.05949942705748576116678424625, 0,
1.05949942705748576116678424625, 2.19778253531805911406355116416, 3.21902491428633574642340557123, 4.05211855016842051337082823228, 4.43642775208533803047423978477, 5.57897628140360803884546521714, 6.31776468449709642497610278420, 6.63963476205521154352416039535, 7.49403317343819764190027743942
