| L(s) = 1 | + (−0.730 − 1.86i)2-s + (−0.0524 − 0.700i)3-s + (−2.93 + 2.72i)4-s + (−8.96 + 6.11i)5-s + (−1.26 + 0.609i)6-s + (15.9 + 9.38i)7-s + (7.20 + 3.47i)8-s + (26.2 − 3.95i)9-s + (17.9 + 12.2i)10-s + (28.3 + 4.26i)11-s + (2.05 + 1.91i)12-s + (−12.0 + 15.1i)13-s + (5.81 − 36.5i)14-s + (4.75 + 5.95i)15-s + (1.19 − 15.9i)16-s + (44.0 − 13.5i)17-s + ⋯ |
| L(s) = 1 | + (−0.258 − 0.658i)2-s + (−0.0100 − 0.134i)3-s + (−0.366 + 0.340i)4-s + (−0.802 + 0.546i)5-s + (−0.0860 + 0.0414i)6-s + (0.861 + 0.506i)7-s + (0.318 + 0.153i)8-s + (0.970 − 0.146i)9-s + (0.567 + 0.386i)10-s + (0.775 + 0.116i)11-s + (0.0495 + 0.0459i)12-s + (−0.257 + 0.323i)13-s + (0.111 − 0.698i)14-s + (0.0817 + 0.102i)15-s + (0.0186 − 0.249i)16-s + (0.628 − 0.194i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00518i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.999 - 0.00518i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.33130 + 0.00345043i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.33130 + 0.00345043i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.730 + 1.86i)T \) |
| 7 | \( 1 + (-15.9 - 9.38i)T \) |
| good | 3 | \( 1 + (0.0524 + 0.700i)T + (-26.6 + 4.02i)T^{2} \) |
| 5 | \( 1 + (8.96 - 6.11i)T + (45.6 - 116. i)T^{2} \) |
| 11 | \( 1 + (-28.3 - 4.26i)T + (1.27e3 + 392. i)T^{2} \) |
| 13 | \( 1 + (12.0 - 15.1i)T + (-488. - 2.14e3i)T^{2} \) |
| 17 | \( 1 + (-44.0 + 13.5i)T + (4.05e3 - 2.76e3i)T^{2} \) |
| 19 | \( 1 + (23.7 - 41.1i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-165. - 51.1i)T + (1.00e4 + 6.85e3i)T^{2} \) |
| 29 | \( 1 + (3.44 - 15.0i)T + (-2.19e4 - 1.05e4i)T^{2} \) |
| 31 | \( 1 + (-32.9 - 57.0i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-108. - 100. i)T + (3.78e3 + 5.05e4i)T^{2} \) |
| 41 | \( 1 + (183. + 88.3i)T + (4.29e4 + 5.38e4i)T^{2} \) |
| 43 | \( 1 + (447. - 215. i)T + (4.95e4 - 6.21e4i)T^{2} \) |
| 47 | \( 1 + (77.3 + 197. i)T + (-7.61e4 + 7.06e4i)T^{2} \) |
| 53 | \( 1 + (-343. + 318. i)T + (1.11e4 - 1.48e5i)T^{2} \) |
| 59 | \( 1 + (234. + 160. i)T + (7.50e4 + 1.91e5i)T^{2} \) |
| 61 | \( 1 + (-153. - 142. i)T + (1.69e4 + 2.26e5i)T^{2} \) |
| 67 | \( 1 + (222. + 384. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + (25.6 + 112. i)T + (-3.22e5 + 1.55e5i)T^{2} \) |
| 73 | \( 1 + (67.7 - 172. i)T + (-2.85e5 - 2.64e5i)T^{2} \) |
| 79 | \( 1 + (-530. + 918. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (923. + 1.15e3i)T + (-1.27e5 + 5.57e5i)T^{2} \) |
| 89 | \( 1 + (-1.54e3 + 232. i)T + (6.73e5 - 2.07e5i)T^{2} \) |
| 97 | \( 1 + 172.T + 9.12e5T^{2} \) |
| 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
−13.22020982916418097169371582053, −11.98537232513682435364867792454, −11.54220816769342166085675902760, −10.32730336278589028562820988805, −9.140438391572317621221086802659, −7.898022341200120495056834730213, −6.88357851159145317172103134916, −4.82044108951671083050316977568, −3.46662249851214063432920647814, −1.53451735924745342641667253114,
1.03126920734964336521692199913, 4.09030809423003381304071731082, 5.00899732687585281302934965892, 6.87138240965579409878186461977, 7.82326421756611905356381815010, 8.789576119441533725434763288462, 10.10468970776988125836184044897, 11.24465058328760856154162220816, 12.42731564737293323439057615799, 13.53512485846816424440246253053
