| L(s) = 1 | − 1.81·2-s − 0.177·3-s + 1.30·4-s − 5-s + 0.321·6-s + 1.07·7-s + 1.26·8-s − 2.96·9-s + 1.81·10-s − 3.31·11-s − 0.230·12-s − 2.65·13-s − 1.95·14-s + 0.177·15-s − 4.90·16-s − 3.99·17-s + 5.39·18-s − 1.30·20-s − 0.190·21-s + 6.01·22-s + 1.75·23-s − 0.224·24-s + 25-s + 4.82·26-s + 1.05·27-s + 1.39·28-s − 2.25·29-s + ⋯ |
| L(s) = 1 | − 1.28·2-s − 0.102·3-s + 0.650·4-s − 0.447·5-s + 0.131·6-s + 0.405·7-s + 0.448·8-s − 0.989·9-s + 0.574·10-s − 0.998·11-s − 0.0665·12-s − 0.737·13-s − 0.521·14-s + 0.0457·15-s − 1.22·16-s − 0.970·17-s + 1.27·18-s − 0.291·20-s − 0.0414·21-s + 1.28·22-s + 0.365·23-s − 0.0458·24-s + 0.200·25-s + 0.947·26-s + 0.203·27-s + 0.264·28-s − 0.418·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3936466532\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3936466532\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 1.81T + 2T^{2} \) |
| 3 | \( 1 + 0.177T + 3T^{2} \) |
| 7 | \( 1 - 1.07T + 7T^{2} \) |
| 11 | \( 1 + 3.31T + 11T^{2} \) |
| 13 | \( 1 + 2.65T + 13T^{2} \) |
| 17 | \( 1 + 3.99T + 17T^{2} \) |
| 23 | \( 1 - 1.75T + 23T^{2} \) |
| 29 | \( 1 + 2.25T + 29T^{2} \) |
| 31 | \( 1 - 8.05T + 31T^{2} \) |
| 37 | \( 1 - 5.64T + 37T^{2} \) |
| 41 | \( 1 - 0.896T + 41T^{2} \) |
| 43 | \( 1 + 8.51T + 43T^{2} \) |
| 47 | \( 1 + 6.29T + 47T^{2} \) |
| 53 | \( 1 - 3.39T + 53T^{2} \) |
| 59 | \( 1 - 1.48T + 59T^{2} \) |
| 61 | \( 1 - 13.7T + 61T^{2} \) |
| 67 | \( 1 + 12.8T + 67T^{2} \) |
| 71 | \( 1 + 11.0T + 71T^{2} \) |
| 73 | \( 1 + 2.61T + 73T^{2} \) |
| 79 | \( 1 - 6.57T + 79T^{2} \) |
| 83 | \( 1 - 2.69T + 83T^{2} \) |
| 89 | \( 1 + 0.789T + 89T^{2} \) |
| 97 | \( 1 - 18.7T + 97T^{2} \) |
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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
−9.131308083998533132204883904511, −8.428770466866069853008530787823, −7.973756254674507788315457183843, −7.23757177249961355937562447087, −6.31071382239566970232217201474, −5.09034095350901452328075902421, −4.52561323788560206485998947875, −2.99472504558902527579870838620, −2.07321737907890338102367460069, −0.49911373524436470861405824914,
0.49911373524436470861405824914, 2.07321737907890338102367460069, 2.99472504558902527579870838620, 4.52561323788560206485998947875, 5.09034095350901452328075902421, 6.31071382239566970232217201474, 7.23757177249961355937562447087, 7.973756254674507788315457183843, 8.428770466866069853008530787823, 9.131308083998533132204883904511
