| L(s) = 1 | − 1.90·2-s − 3-s + 1.62·4-s + 1.90·6-s + 4.42·7-s + 0.719·8-s + 9-s + 11-s − 1.62·12-s + 0.622·13-s − 8.42·14-s − 4.61·16-s + 5.18·17-s − 1.90·18-s + 7.05·19-s − 4.42·21-s − 1.90·22-s − 8.85·23-s − 0.719·24-s − 1.18·26-s − 27-s + 7.18·28-s − 7.80·29-s + 2.75·31-s + 7.34·32-s − 33-s − 9.86·34-s + ⋯ |
| L(s) = 1 | − 1.34·2-s − 0.577·3-s + 0.811·4-s + 0.776·6-s + 1.67·7-s + 0.254·8-s + 0.333·9-s + 0.301·11-s − 0.468·12-s + 0.172·13-s − 2.25·14-s − 1.15·16-s + 1.25·17-s − 0.448·18-s + 1.61·19-s − 0.966·21-s − 0.405·22-s − 1.84·23-s − 0.146·24-s − 0.232·26-s − 0.192·27-s + 1.35·28-s − 1.44·29-s + 0.494·31-s + 1.29·32-s − 0.174·33-s − 1.69·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8169526999\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8169526999\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 + 1.90T + 2T^{2} \) |
| 7 | \( 1 - 4.42T + 7T^{2} \) |
| 13 | \( 1 - 0.622T + 13T^{2} \) |
| 17 | \( 1 - 5.18T + 17T^{2} \) |
| 19 | \( 1 - 7.05T + 19T^{2} \) |
| 23 | \( 1 + 8.85T + 23T^{2} \) |
| 29 | \( 1 + 7.80T + 29T^{2} \) |
| 31 | \( 1 - 2.75T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + 0.193T + 41T^{2} \) |
| 43 | \( 1 + 5.67T + 43T^{2} \) |
| 47 | \( 1 - 2.75T + 47T^{2} \) |
| 53 | \( 1 - 10.8T + 53T^{2} \) |
| 59 | \( 1 + 4.85T + 59T^{2} \) |
| 61 | \( 1 - 6.85T + 61T^{2} \) |
| 67 | \( 1 - 1.24T + 67T^{2} \) |
| 71 | \( 1 - 2.75T + 71T^{2} \) |
| 73 | \( 1 + 4.23T + 73T^{2} \) |
| 79 | \( 1 - 8.56T + 79T^{2} \) |
| 83 | \( 1 + 0.133T + 83T^{2} \) |
| 89 | \( 1 - 5.61T + 89T^{2} \) |
| 97 | \( 1 + 7.24T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.09655303064597242717470941601, −9.509729547902967316931157586557, −8.409578853511103745454259503108, −7.80148911629081451534797706316, −7.26252456607141353211515782971, −5.82175593387592209920033989527, −5.04109463830870922650412143532, −3.90153366285414162564750922574, −1.91027492340035911089610934674, −1.01429119232244115145294490384,
1.01429119232244115145294490384, 1.91027492340035911089610934674, 3.90153366285414162564750922574, 5.04109463830870922650412143532, 5.82175593387592209920033989527, 7.26252456607141353211515782971, 7.80148911629081451534797706316, 8.409578853511103745454259503108, 9.509729547902967316931157586557, 10.09655303064597242717470941601
