| L(s) = 1 | + (0.125 − 0.992i)2-s + (−0.597 + 0.493i)3-s + (−0.968 − 0.248i)4-s + (1.15 + 1.91i)5-s + (0.415 + 0.654i)6-s + (−0.100 + 0.0326i)7-s + (−0.368 + 0.929i)8-s + (−0.449 + 2.35i)9-s + (2.04 − 0.904i)10-s + (4.93 + 0.623i)11-s + (0.701 − 0.329i)12-s + (0.718 + 0.137i)13-s + (0.0197 + 0.103i)14-s + (−1.63 − 0.573i)15-s + (0.876 + 0.481i)16-s + (0.105 + 0.409i)17-s + ⋯ |
| L(s) = 1 | + (0.0886 − 0.701i)2-s + (−0.344 + 0.285i)3-s + (−0.484 − 0.124i)4-s + (0.516 + 0.856i)5-s + (0.169 + 0.267i)6-s + (−0.0379 + 0.0123i)7-s + (−0.130 + 0.328i)8-s + (−0.149 + 0.785i)9-s + (0.646 − 0.286i)10-s + (1.48 + 0.188i)11-s + (0.202 − 0.0952i)12-s + (0.199 + 0.0380i)13-s + (0.00529 + 0.0277i)14-s + (−0.422 − 0.148i)15-s + (0.219 + 0.120i)16-s + (0.0255 + 0.0993i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 - 0.217i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.976 - 0.217i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.22954 + 0.135302i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.22954 + 0.135302i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.125 + 0.992i)T \) |
| 5 | \( 1 + (-1.15 - 1.91i)T \) |
| good | 3 | \( 1 + (0.597 - 0.493i)T + (0.562 - 2.94i)T^{2} \) |
| 7 | \( 1 + (0.100 - 0.0326i)T + (5.66 - 4.11i)T^{2} \) |
| 11 | \( 1 + (-4.93 - 0.623i)T + (10.6 + 2.73i)T^{2} \) |
| 13 | \( 1 + (-0.718 - 0.137i)T + (12.0 + 4.78i)T^{2} \) |
| 17 | \( 1 + (-0.105 - 0.409i)T + (-14.8 + 8.18i)T^{2} \) |
| 19 | \( 1 + (-0.380 + 0.459i)T + (-3.56 - 18.6i)T^{2} \) |
| 23 | \( 1 + (-0.686 + 0.730i)T + (-1.44 - 22.9i)T^{2} \) |
| 29 | \( 1 + (0.0793 + 1.26i)T + (-28.7 + 3.63i)T^{2} \) |
| 31 | \( 1 + (-3.56 + 0.916i)T + (27.1 - 14.9i)T^{2} \) |
| 37 | \( 1 + (0.731 - 1.33i)T + (-19.8 - 31.2i)T^{2} \) |
| 41 | \( 1 + (1.74 - 1.63i)T + (2.57 - 40.9i)T^{2} \) |
| 43 | \( 1 + (2.84 + 3.92i)T + (-13.2 + 40.8i)T^{2} \) |
| 47 | \( 1 + (2.97 + 7.52i)T + (-34.2 + 32.1i)T^{2} \) |
| 53 | \( 1 + (10.7 + 6.79i)T + (22.5 + 47.9i)T^{2} \) |
| 59 | \( 1 + (-0.941 - 2.00i)T + (-37.6 + 45.4i)T^{2} \) |
| 61 | \( 1 + (-6.54 - 6.14i)T + (3.83 + 60.8i)T^{2} \) |
| 67 | \( 1 + (-3.20 - 0.201i)T + (66.4 + 8.39i)T^{2} \) |
| 71 | \( 1 + (-9.53 + 3.77i)T + (51.7 - 48.6i)T^{2} \) |
| 73 | \( 1 + (-0.398 - 0.187i)T + (46.5 + 56.2i)T^{2} \) |
| 79 | \( 1 + (-1.31 - 1.58i)T + (-14.8 + 77.6i)T^{2} \) |
| 83 | \( 1 + (8.23 + 6.81i)T + (15.5 + 81.5i)T^{2} \) |
| 89 | \( 1 + (-6.14 + 13.0i)T + (-56.7 - 68.5i)T^{2} \) |
| 97 | \( 1 + (-0.511 + 0.0321i)T + (96.2 - 12.1i)T^{2} \) |
| 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
−11.71955711308355832962524769911, −11.24415297125271620693266283350, −10.23955529518265418823414093659, −9.593884183901340474745088136000, −8.394861268042675616881851482081, −6.92186301437176738248688851643, −5.94028309406888368114605710992, −4.66073163887619418363942846417, −3.37857007899036476901026172303, −1.90323632732211616285866687780,
1.18300650119371418005556349599, 3.67925529642206454951651504726, 4.96774865530391537753090118867, 6.13918259068796869313235188838, 6.68609405510812734001218773387, 8.158368367673427357577964708612, 9.123810818709810417849431986186, 9.686313658138052288698502474912, 11.32578264937985674426680405012, 12.20949090391698694288151055695
