| L(s) = 1 | + (−1.99 − 0.142i)2-s + (−0.143 − 0.661i)3-s + (3.95 + 0.569i)4-s + (10.6 + 3.51i)5-s + (0.192 + 1.33i)6-s + (−7.16 − 13.1i)7-s + (−7.81 − 1.70i)8-s + (24.1 − 11.0i)9-s + (−20.6 − 8.53i)10-s + (24.5 + 21.2i)11-s + (−0.193 − 2.69i)12-s + (−36.6 − 20.0i)13-s + (12.4 + 27.1i)14-s + (0.798 − 7.52i)15-s + (15.3 + 4.50i)16-s + (9.48 − 12.6i)17-s + ⋯ |
| L(s) = 1 | + (−0.705 − 0.0504i)2-s + (−0.0276 − 0.127i)3-s + (0.494 + 0.0711i)4-s + (0.949 + 0.314i)5-s + (0.0131 + 0.0911i)6-s + (−0.386 − 0.708i)7-s + (−0.345 − 0.0751i)8-s + (0.894 − 0.408i)9-s + (−0.653 − 0.269i)10-s + (0.672 + 0.582i)11-s + (−0.00464 − 0.0649i)12-s + (−0.782 − 0.427i)13-s + (0.237 + 0.518i)14-s + (0.0137 − 0.129i)15-s + (0.239 + 0.0704i)16-s + (0.135 − 0.180i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.745 + 0.666i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.745 + 0.666i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.42621 - 0.544244i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.42621 - 0.544244i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.99 + 0.142i)T \) |
| 5 | \( 1 + (-10.6 - 3.51i)T \) |
| 23 | \( 1 + (10.4 - 109. i)T \) |
| good | 3 | \( 1 + (0.143 + 0.661i)T + (-24.5 + 11.2i)T^{2} \) |
| 7 | \( 1 + (7.16 + 13.1i)T + (-185. + 288. i)T^{2} \) |
| 11 | \( 1 + (-24.5 - 21.2i)T + (189. + 1.31e3i)T^{2} \) |
| 13 | \( 1 + (36.6 + 20.0i)T + (1.18e3 + 1.84e3i)T^{2} \) |
| 17 | \( 1 + (-9.48 + 12.6i)T + (-1.38e3 - 4.71e3i)T^{2} \) |
| 19 | \( 1 + (-8.10 + 56.3i)T + (-6.58e3 - 1.93e3i)T^{2} \) |
| 29 | \( 1 + (-34.0 + 4.89i)T + (2.34e4 - 6.87e3i)T^{2} \) |
| 31 | \( 1 + (-79.8 + 51.3i)T + (1.23e4 - 2.70e4i)T^{2} \) |
| 37 | \( 1 + (-149. + 399. i)T + (-3.82e4 - 3.31e4i)T^{2} \) |
| 41 | \( 1 + (-101. + 222. i)T + (-4.51e4 - 5.20e4i)T^{2} \) |
| 43 | \( 1 + (-9.81 + 2.13i)T + (7.23e4 - 3.30e4i)T^{2} \) |
| 47 | \( 1 + (-180. + 180. i)T - 1.03e5iT^{2} \) |
| 53 | \( 1 + (-612. + 334. i)T + (8.04e4 - 1.25e5i)T^{2} \) |
| 59 | \( 1 + (-21.1 - 71.9i)T + (-1.72e5 + 1.11e5i)T^{2} \) |
| 61 | \( 1 + (-69.9 - 108. i)T + (-9.42e4 + 2.06e5i)T^{2} \) |
| 67 | \( 1 + (65.5 - 916. i)T + (-2.97e5 - 4.28e4i)T^{2} \) |
| 71 | \( 1 + (-372. - 429. i)T + (-5.09e4 + 3.54e5i)T^{2} \) |
| 73 | \( 1 + (437. - 327. i)T + (1.09e5 - 3.73e5i)T^{2} \) |
| 79 | \( 1 + (307. - 90.1i)T + (4.14e5 - 2.66e5i)T^{2} \) |
| 83 | \( 1 + (1.27e3 + 475. i)T + (4.32e5 + 3.74e5i)T^{2} \) |
| 89 | \( 1 + (-211. - 135. i)T + (2.92e5 + 6.41e5i)T^{2} \) |
| 97 | \( 1 + (605. - 225. i)T + (6.89e5 - 5.97e5i)T^{2} \) |
| 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
−11.53215996640052655630989196015, −10.24670408974756517701879002354, −9.858479838684835838044337411912, −9.025811231711836370911838588107, −7.23433122218392814239707240651, −7.03798595158407587241866910722, −5.64725589226196295810098659437, −3.98720867255597743258373739403, −2.35299682551777286075433977658, −0.918180546203569508188224984413,
1.33035378049078469869025023391, 2.64863650272454801899173211828, 4.57490735010140443210279295291, 5.92324124802308828095638452089, 6.72708979278850568275581316204, 8.117558507729193518690799698604, 9.126836678256612666868887624537, 9.811539368367125405310309044991, 10.56226999021794028230504476171, 11.92784889045601166884080944462
