| L(s) = 1 | + (−3.72 + 3.62i)3-s + (−6.50 − 2.36i)5-s + (1.34 − 7.63i)7-s + (0.689 − 26.9i)9-s + (13.7 − 5.00i)11-s + (40.8 − 34.3i)13-s + (32.7 − 14.7i)15-s + (−29.6 − 51.2i)17-s + (46.8 − 81.2i)19-s + (22.6 + 33.3i)21-s + (4.51 + 25.6i)23-s + (−59.0 − 49.5i)25-s + (95.3 + 102. i)27-s + (−172. − 144. i)29-s + (4.27 + 24.2i)31-s + ⋯ |
| L(s) = 1 | + (−0.716 + 0.698i)3-s + (−0.581 − 0.211i)5-s + (0.0727 − 0.412i)7-s + (0.0255 − 0.999i)9-s + (0.376 − 0.137i)11-s + (0.872 − 0.731i)13-s + (0.564 − 0.254i)15-s + (−0.422 − 0.731i)17-s + (0.566 − 0.980i)19-s + (0.235 + 0.346i)21-s + (0.0409 + 0.232i)23-s + (−0.472 − 0.396i)25-s + (0.679 + 0.733i)27-s + (−1.10 − 0.926i)29-s + (0.0247 + 0.140i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.333 + 0.942i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.333 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.718221 - 0.507854i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.718221 - 0.507854i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (3.72 - 3.62i)T \) |
| good | 5 | \( 1 + (6.50 + 2.36i)T + (95.7 + 80.3i)T^{2} \) |
| 7 | \( 1 + (-1.34 + 7.63i)T + (-322. - 117. i)T^{2} \) |
| 11 | \( 1 + (-13.7 + 5.00i)T + (1.01e3 - 855. i)T^{2} \) |
| 13 | \( 1 + (-40.8 + 34.3i)T + (381. - 2.16e3i)T^{2} \) |
| 17 | \( 1 + (29.6 + 51.2i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-46.8 + 81.2i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-4.51 - 25.6i)T + (-1.14e4 + 4.16e3i)T^{2} \) |
| 29 | \( 1 + (172. + 144. i)T + (4.23e3 + 2.40e4i)T^{2} \) |
| 31 | \( 1 + (-4.27 - 24.2i)T + (-2.79e4 + 1.01e4i)T^{2} \) |
| 37 | \( 1 + (38.3 + 66.4i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (305. - 256. i)T + (1.19e4 - 6.78e4i)T^{2} \) |
| 43 | \( 1 + (-282. + 102. i)T + (6.09e4 - 5.11e4i)T^{2} \) |
| 47 | \( 1 + (3.08 - 17.4i)T + (-9.75e4 - 3.55e4i)T^{2} \) |
| 53 | \( 1 + 242.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (211. + 76.9i)T + (1.57e5 + 1.32e5i)T^{2} \) |
| 61 | \( 1 + (-81.1 + 460. i)T + (-2.13e5 - 7.76e4i)T^{2} \) |
| 67 | \( 1 + (-333. + 279. i)T + (5.22e4 - 2.96e5i)T^{2} \) |
| 71 | \( 1 + (-277. - 480. i)T + (-1.78e5 + 3.09e5i)T^{2} \) |
| 73 | \( 1 + (226. - 393. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (537. + 450. i)T + (8.56e4 + 4.85e5i)T^{2} \) |
| 83 | \( 1 + (427. + 358. i)T + (9.92e4 + 5.63e5i)T^{2} \) |
| 89 | \( 1 + (-578. + 1.00e3i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-1.38e3 + 505. i)T + (6.99e5 - 5.86e5i)T^{2} \) |
| 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
−12.93651364353374894488171710232, −11.61922115390358105109251698312, −11.13654126955194545749244869776, −9.902062046802510277726902709629, −8.809350464987761136162540092521, −7.39718200628900198897561485525, −6.02016956744534829364359976125, −4.70604744651920877384543837873, −3.53770802671748573988770919104, −0.55850074048595720593879752653,
1.66377507037202744677706867294, 3.88176181053436686523725481949, 5.57232620742120670882454210851, 6.66240482032260888401930060711, 7.77483252350492216254512429319, 8.957981900978280661384179605667, 10.58249119027995447020106978497, 11.50338447604072334731295038978, 12.20840842272240961238685592791, 13.27661985009120804628524386647
