| L(s) = 1 | − 3.23·5-s − 4.47·11-s + 1.23·13-s + 2.47·17-s + 19-s − 2.76·23-s + 5.47·25-s + 6·29-s + 7.23·31-s + 5.23·37-s + 2·41-s + 8.94·43-s + 11.7·47-s − 7·49-s − 8.47·53-s + 14.4·55-s − 14.4·59-s − 4.47·61-s − 4.00·65-s − 5.52·67-s − 5.52·71-s − 8.47·73-s + 1.70·79-s − 10·83-s − 8.00·85-s + 6·89-s − 3.23·95-s + ⋯ |
| L(s) = 1 | − 1.44·5-s − 1.34·11-s + 0.342·13-s + 0.599·17-s + 0.229·19-s − 0.576·23-s + 1.09·25-s + 1.11·29-s + 1.29·31-s + 0.860·37-s + 0.312·41-s + 1.36·43-s + 1.70·47-s − 49-s − 1.16·53-s + 1.95·55-s − 1.88·59-s − 0.572·61-s − 0.496·65-s − 0.675·67-s − 0.656·71-s − 0.991·73-s + 0.192·79-s − 1.09·83-s − 0.867·85-s + 0.635·89-s − 0.332·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5472 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 + 3.23T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 4.47T + 11T^{2} \) |
| 13 | \( 1 - 1.23T + 13T^{2} \) |
| 17 | \( 1 - 2.47T + 17T^{2} \) |
| 23 | \( 1 + 2.76T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 7.23T + 31T^{2} \) |
| 37 | \( 1 - 5.23T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 8.94T + 43T^{2} \) |
| 47 | \( 1 - 11.7T + 47T^{2} \) |
| 53 | \( 1 + 8.47T + 53T^{2} \) |
| 59 | \( 1 + 14.4T + 59T^{2} \) |
| 61 | \( 1 + 4.47T + 61T^{2} \) |
| 67 | \( 1 + 5.52T + 67T^{2} \) |
| 71 | \( 1 + 5.52T + 71T^{2} \) |
| 73 | \( 1 + 8.47T + 73T^{2} \) |
| 79 | \( 1 - 1.70T + 79T^{2} \) |
| 83 | \( 1 + 10T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 7.52T + 97T^{2} \) |
| 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
−7.80928570052790728640746432761, −7.43568093337079432181038696445, −6.34830922365012256644569095863, −5.67191723100152524650022648967, −4.63421523178936035622057976531, −4.25317277142957090004965996122, −3.16715389167627899620088904104, −2.66840955514093036617539217851, −1.11020688743855259061933777026, 0,
1.11020688743855259061933777026, 2.66840955514093036617539217851, 3.16715389167627899620088904104, 4.25317277142957090004965996122, 4.63421523178936035622057976531, 5.67191723100152524650022648967, 6.34830922365012256644569095863, 7.43568093337079432181038696445, 7.80928570052790728640746432761
