| L(s) = 1 | − 3.23·5-s − 7-s + 0.763·11-s − 5.23·13-s − 2.76·17-s + 19-s − 1.23·23-s + 5.47·25-s + 0.472·29-s − 8.47·31-s + 3.23·35-s − 8.94·37-s − 2·41-s − 4·43-s − 2.47·47-s + 49-s − 8.47·53-s − 2.47·55-s + 8·59-s − 0.472·61-s + 16.9·65-s + 5.23·67-s + 10.4·71-s − 4.47·73-s − 0.763·77-s + 2.29·79-s − 2·83-s + ⋯ |
| L(s) = 1 | − 1.44·5-s − 0.377·7-s + 0.230·11-s − 1.45·13-s − 0.670·17-s + 0.229·19-s − 0.257·23-s + 1.09·25-s + 0.0876·29-s − 1.52·31-s + 0.546·35-s − 1.47·37-s − 0.312·41-s − 0.609·43-s − 0.360·47-s + 0.142·49-s − 1.16·53-s − 0.333·55-s + 1.04·59-s − 0.0604·61-s + 2.10·65-s + 0.639·67-s + 1.24·71-s − 0.523·73-s − 0.0870·77-s + 0.257·79-s − 0.219·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3458866554\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3458866554\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 + 3.23T + 5T^{2} \) |
| 11 | \( 1 - 0.763T + 11T^{2} \) |
| 13 | \( 1 + 5.23T + 13T^{2} \) |
| 17 | \( 1 + 2.76T + 17T^{2} \) |
| 23 | \( 1 + 1.23T + 23T^{2} \) |
| 29 | \( 1 - 0.472T + 29T^{2} \) |
| 31 | \( 1 + 8.47T + 31T^{2} \) |
| 37 | \( 1 + 8.94T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + 2.47T + 47T^{2} \) |
| 53 | \( 1 + 8.47T + 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 + 0.472T + 61T^{2} \) |
| 67 | \( 1 - 5.23T + 67T^{2} \) |
| 71 | \( 1 - 10.4T + 71T^{2} \) |
| 73 | \( 1 + 4.47T + 73T^{2} \) |
| 79 | \( 1 - 2.29T + 79T^{2} \) |
| 83 | \( 1 + 2T + 83T^{2} \) |
| 89 | \( 1 + 6.94T + 89T^{2} \) |
| 97 | \( 1 + 3.23T + 97T^{2} \) |
| 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
−7.69247814917445102402354737473, −6.95761701017399195808708798476, −6.69962296376422786067278444002, −5.44107036732067245304075133221, −4.91549410761598101225538044907, −4.07253397121681352589706421983, −3.56455118970851291484025486510, −2.73826941560852841838158480692, −1.74955773910235877915909269219, −0.26877759296438580179967345213,
0.26877759296438580179967345213, 1.74955773910235877915909269219, 2.73826941560852841838158480692, 3.56455118970851291484025486510, 4.07253397121681352589706421983, 4.91549410761598101225538044907, 5.44107036732067245304075133221, 6.69962296376422786067278444002, 6.95761701017399195808708798476, 7.69247814917445102402354737473
