| L(s) = 1 | + 0.585·7-s − 5.41·11-s + 0.585i·13-s − 6.82·17-s − 0.828i·23-s − 6i·29-s + 10.4i·31-s + 5.07i·37-s − 3.07i·41-s + 1.17·43-s − 5.65i·47-s − 6.65·49-s + 6.82·53-s + 9.41·59-s + 7.17·61-s + ⋯ |
| L(s) = 1 | + 0.221·7-s − 1.63·11-s + 0.162i·13-s − 1.65·17-s − 0.172i·23-s − 1.11i·29-s + 1.88i·31-s + 0.833i·37-s − 0.479i·41-s + 0.178·43-s − 0.825i·47-s − 0.950·49-s + 0.937·53-s + 1.22·59-s + 0.918·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.289i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.957 + 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.281678423\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.281678423\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 - 0.585T + 7T^{2} \) |
| 11 | \( 1 + 5.41T + 11T^{2} \) |
| 13 | \( 1 - 0.585iT - 13T^{2} \) |
| 17 | \( 1 + 6.82T + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 0.828iT - 23T^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 - 10.4iT - 31T^{2} \) |
| 37 | \( 1 - 5.07iT - 37T^{2} \) |
| 41 | \( 1 + 3.07iT - 41T^{2} \) |
| 43 | \( 1 - 1.17T + 43T^{2} \) |
| 47 | \( 1 + 5.65iT - 47T^{2} \) |
| 53 | \( 1 - 6.82T + 53T^{2} \) |
| 59 | \( 1 - 9.41T + 59T^{2} \) |
| 61 | \( 1 - 7.17T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 - 5.65T + 71T^{2} \) |
| 73 | \( 1 + 6.48iT - 73T^{2} \) |
| 79 | \( 1 + 2.48iT - 79T^{2} \) |
| 83 | \( 1 - 14.8iT - 83T^{2} \) |
| 89 | \( 1 + 4.24iT - 89T^{2} \) |
| 97 | \( 1 + 14.4iT - 97T^{2} \) |
| 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
−8.044871342928529894507803974203, −7.08127766779497596149051494082, −6.66688422919726667254807877996, −5.68739731605859771000814887824, −5.01461706523697598330738113264, −4.47503098014702205048013770220, −3.48020753664247960777942961425, −2.53490894016201736268093226905, −1.96752291682562092640747186340, −0.48950863938331367916346157366,
0.60578696835215163020183016291, 2.11156603522544131714565743801, 2.53911773803423443614873540490, 3.61867248576344371556626724078, 4.47349032630588005411398639298, 5.13480640253248385313256220144, 5.77011518204514402864566489688, 6.60281664335009615886767768884, 7.34789794402558911426945075765, 7.941564694739815592162578697394
