| L(s) = 1 | − 5-s + 1.43·7-s + 1.35·11-s − 5.52·13-s − 4.82·17-s − 0.648·19-s + 8.90·23-s + 25-s + 7.17·29-s − 4.64·31-s − 1.43·35-s + 1.35·37-s − 0.351·41-s + 4.82·43-s + 9.49·47-s − 4.93·49-s + 8.17·53-s − 1.35·55-s − 1.46·59-s + 6.69·61-s + 5.52·65-s + 12.4·67-s − 2.22·71-s − 4.34·73-s + 1.94·77-s − 13.0·79-s + 5.26·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.543·7-s + 0.407·11-s − 1.53·13-s − 1.16·17-s − 0.148·19-s + 1.85·23-s + 0.200·25-s + 1.33·29-s − 0.834·31-s − 0.243·35-s + 0.222·37-s − 0.0549·41-s + 0.735·43-s + 1.38·47-s − 0.704·49-s + 1.12·53-s − 0.182·55-s − 0.191·59-s + 0.857·61-s + 0.685·65-s + 1.51·67-s − 0.264·71-s − 0.508·73-s + 0.221·77-s − 1.46·79-s + 0.577·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.606214905\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.606214905\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| good | 7 | \( 1 - 1.43T + 7T^{2} \) |
| 11 | \( 1 - 1.35T + 11T^{2} \) |
| 13 | \( 1 + 5.52T + 13T^{2} \) |
| 17 | \( 1 + 4.82T + 17T^{2} \) |
| 19 | \( 1 + 0.648T + 19T^{2} \) |
| 23 | \( 1 - 8.90T + 23T^{2} \) |
| 29 | \( 1 - 7.17T + 29T^{2} \) |
| 31 | \( 1 + 4.64T + 31T^{2} \) |
| 37 | \( 1 - 1.35T + 37T^{2} \) |
| 41 | \( 1 + 0.351T + 41T^{2} \) |
| 43 | \( 1 - 4.82T + 43T^{2} \) |
| 47 | \( 1 - 9.49T + 47T^{2} \) |
| 53 | \( 1 - 8.17T + 53T^{2} \) |
| 59 | \( 1 + 1.46T + 59T^{2} \) |
| 61 | \( 1 - 6.69T + 61T^{2} \) |
| 67 | \( 1 - 12.4T + 67T^{2} \) |
| 71 | \( 1 + 2.22T + 71T^{2} \) |
| 73 | \( 1 + 4.34T + 73T^{2} \) |
| 79 | \( 1 + 13.0T + 79T^{2} \) |
| 83 | \( 1 - 5.26T + 83T^{2} \) |
| 89 | \( 1 - 11T + 89T^{2} \) |
| 97 | \( 1 - 17.5T + 97T^{2} \) |
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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
−8.782125424327505030988421337054, −7.84507476365962997418413227615, −7.11700818745163406138948595023, −6.67350916231201098957082043365, −5.43199846600304346592408594146, −4.74578755427791903554201022641, −4.15486022797366647899467277376, −2.94295785311676207777245124738, −2.14446588557025644013056409854, −0.75466189941054143120195042745,
0.75466189941054143120195042745, 2.14446588557025644013056409854, 2.94295785311676207777245124738, 4.15486022797366647899467277376, 4.74578755427791903554201022641, 5.43199846600304346592408594146, 6.67350916231201098957082043365, 7.11700818745163406138948595023, 7.84507476365962997418413227615, 8.782125424327505030988421337054
