| 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 | \( 1 + (605. + 225. i)T + (6.89e5 + 5.97e5i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.92784889045601166884080944462, −10.56226999021794028230504476171, −9.811539368367125405310309044991, −9.126836678256612666868887624537, −8.117558507729193518690799698604, −6.72708979278850568275581316204, −5.92324124802308828095638452089, −4.57490735010140443210279295291, −2.64863650272454801899173211828, −1.33035378049078469869025023391,
0.918180546203569508188224984413, 2.35299682551777286075433977658, 3.98720867255597743258373739403, 5.64725589226196295810098659437, 7.03798595158407587241866910722, 7.23433122218392814239707240651, 9.025811231711836370911838588107, 9.858479838684835838044337411912, 10.24670408974756517701879002354, 11.53215996640052655630989196015
