| L(s) = 1 | + (3.12 + 4.29i)2-s + (5.96 + 1.93i)3-s + (−6.24 + 19.2i)4-s + (10.2 + 31.6i)6-s − 9.63i·7-s + (−61.6 + 20.0i)8-s + (9.96 + 7.24i)9-s + (19.6 − 14.3i)11-s + (−74.5 + 102. i)12-s + (25.9 − 35.7i)13-s + (41.4 − 30.0i)14-s + (−147. − 107. i)16-s + (−1.75 + 0.568i)17-s + 65.4i·18-s + (22.1 + 68.2i)19-s + ⋯ |
| L(s) = 1 | + (1.10 + 1.51i)2-s + (1.14 + 0.372i)3-s + (−0.780 + 2.40i)4-s + (0.700 + 2.15i)6-s − 0.520i·7-s + (−2.72 + 0.885i)8-s + (0.369 + 0.268i)9-s + (0.539 − 0.392i)11-s + (−1.79 + 2.46i)12-s + (0.554 − 0.762i)13-s + (0.790 − 0.574i)14-s + (−2.31 − 1.67i)16-s + (−0.0249 + 0.00811i)17-s + 0.857i·18-s + (0.267 + 0.824i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.786 - 0.617i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.786 - 0.617i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.20695 + 3.49507i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.20695 + 3.49507i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 + (-3.12 - 4.29i)T + (-2.47 + 7.60i)T^{2} \) |
| 3 | \( 1 + (-5.96 - 1.93i)T + (21.8 + 15.8i)T^{2} \) |
| 7 | \( 1 + 9.63iT - 343T^{2} \) |
| 11 | \( 1 + (-19.6 + 14.3i)T + (411. - 1.26e3i)T^{2} \) |
| 13 | \( 1 + (-25.9 + 35.7i)T + (-678. - 2.08e3i)T^{2} \) |
| 17 | \( 1 + (1.75 - 0.568i)T + (3.97e3 - 2.88e3i)T^{2} \) |
| 19 | \( 1 + (-22.1 - 68.2i)T + (-5.54e3 + 4.03e3i)T^{2} \) |
| 23 | \( 1 + (-41.2 - 56.7i)T + (-3.75e3 + 1.15e4i)T^{2} \) |
| 29 | \( 1 + (-31.0 + 95.6i)T + (-1.97e4 - 1.43e4i)T^{2} \) |
| 31 | \( 1 + (51.1 + 157. i)T + (-2.41e4 + 1.75e4i)T^{2} \) |
| 37 | \( 1 + (229. - 316. i)T + (-1.56e4 - 4.81e4i)T^{2} \) |
| 41 | \( 1 + (2.89 + 2.10i)T + (2.12e4 + 6.55e4i)T^{2} \) |
| 43 | \( 1 + 9.64iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (515. + 167. i)T + (8.39e4 + 6.10e4i)T^{2} \) |
| 53 | \( 1 + (154. + 50.2i)T + (1.20e5 + 8.75e4i)T^{2} \) |
| 59 | \( 1 + (-134. - 97.6i)T + (6.34e4 + 1.95e5i)T^{2} \) |
| 61 | \( 1 + (-692. + 502. i)T + (7.01e4 - 2.15e5i)T^{2} \) |
| 67 | \( 1 + (-671. + 218. i)T + (2.43e5 - 1.76e5i)T^{2} \) |
| 71 | \( 1 + (209. - 644. i)T + (-2.89e5 - 2.10e5i)T^{2} \) |
| 73 | \( 1 + (28.4 + 39.1i)T + (-1.20e5 + 3.69e5i)T^{2} \) |
| 79 | \( 1 + (145. - 448. i)T + (-3.98e5 - 2.89e5i)T^{2} \) |
| 83 | \( 1 + (572. - 185. i)T + (4.62e5 - 3.36e5i)T^{2} \) |
| 89 | \( 1 + (-558. + 406. i)T + (2.17e5 - 6.70e5i)T^{2} \) |
| 97 | \( 1 + (585. + 190. i)T + (7.38e5 + 5.36e5i)T^{2} \) |
| 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
−13.70475047906009940005599354914, −12.98797734253873016026827356665, −11.61859768118314118418416475424, −9.788495819326729588582408006207, −8.496266537184486806344099205666, −7.945555727378526252040719653876, −6.68038101160613246097429941160, −5.48823779368131112310513278443, −3.98822127438334949576992457217, −3.27354321689475131427746351491,
1.60533932753509124927822306906, 2.70362074289415435153519405033, 3.82912426266088561053983294640, 5.18674105443833975669465510130, 6.78900826137997148229192316497, 8.759298492229620547734280866433, 9.361923868011979554702464062288, 10.74768879020921583498719969044, 11.70086363844434253870767578070, 12.62850703672806166682873714459
