| L(s) = 1 | + (−3.80 + 1.23i)2-s + (−6.76 − 9.31i)3-s + (12.9 − 9.40i)4-s + (−7.78 + 55.3i)5-s + (37.2 + 27.0i)6-s + 51.2i·7-s + (−37.6 + 51.7i)8-s + (34.1 − 105. i)9-s + (−38.8 − 220. i)10-s + (−166. − 513. i)11-s + (−175. − 56.9i)12-s + (765. + 248. i)13-s + (−63.3 − 195. i)14-s + (568. − 302. i)15-s + (79.1 − 243. i)16-s + (1.27e3 − 1.75e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.672 + 0.218i)2-s + (−0.434 − 0.597i)3-s + (0.404 − 0.293i)4-s + (−0.139 + 0.990i)5-s + (0.422 + 0.306i)6-s + 0.395i·7-s + (−0.207 + 0.286i)8-s + (0.140 − 0.432i)9-s + (−0.122 − 0.696i)10-s + (−0.415 − 1.27i)11-s + (−0.351 − 0.114i)12-s + (1.25 + 0.408i)13-s + (−0.0864 − 0.265i)14-s + (0.651 − 0.346i)15-s + (0.0772 − 0.237i)16-s + (1.06 − 1.47i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.663 + 0.747i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.663 + 0.747i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.886665 - 0.398515i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.886665 - 0.398515i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (3.80 - 1.23i)T \) |
| 5 | \( 1 + (7.78 - 55.3i)T \) |
| good | 3 | \( 1 + (6.76 + 9.31i)T + (-75.0 + 231. i)T^{2} \) |
| 7 | \( 1 - 51.2iT - 1.68e4T^{2} \) |
| 11 | \( 1 + (166. + 513. i)T + (-1.30e5 + 9.46e4i)T^{2} \) |
| 13 | \( 1 + (-765. - 248. i)T + (3.00e5 + 2.18e5i)T^{2} \) |
| 17 | \( 1 + (-1.27e3 + 1.75e3i)T + (-4.38e5 - 1.35e6i)T^{2} \) |
| 19 | \( 1 + (369. + 268. i)T + (7.65e5 + 2.35e6i)T^{2} \) |
| 23 | \( 1 + (-2.81e3 + 916. i)T + (5.20e6 - 3.78e6i)T^{2} \) |
| 29 | \( 1 + (-3.78e3 + 2.75e3i)T + (6.33e6 - 1.95e7i)T^{2} \) |
| 31 | \( 1 + (3.84e3 + 2.79e3i)T + (8.84e6 + 2.72e7i)T^{2} \) |
| 37 | \( 1 + (-9.63e3 - 3.13e3i)T + (5.61e7 + 4.07e7i)T^{2} \) |
| 41 | \( 1 + (1.70e3 - 5.24e3i)T + (-9.37e7 - 6.80e7i)T^{2} \) |
| 43 | \( 1 - 1.71e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + (4.28e3 + 5.89e3i)T + (-7.08e7 + 2.18e8i)T^{2} \) |
| 53 | \( 1 + (6.06e3 + 8.34e3i)T + (-1.29e8 + 3.97e8i)T^{2} \) |
| 59 | \( 1 + (-1.42e4 + 4.37e4i)T + (-5.78e8 - 4.20e8i)T^{2} \) |
| 61 | \( 1 + (-1.00e3 - 3.10e3i)T + (-6.83e8 + 4.96e8i)T^{2} \) |
| 67 | \( 1 + (-1.92e4 + 2.64e4i)T + (-4.17e8 - 1.28e9i)T^{2} \) |
| 71 | \( 1 + (6.98e3 - 5.07e3i)T + (5.57e8 - 1.71e9i)T^{2} \) |
| 73 | \( 1 + (5.59e4 - 1.81e4i)T + (1.67e9 - 1.21e9i)T^{2} \) |
| 79 | \( 1 + (-2.01e4 + 1.46e4i)T + (9.50e8 - 2.92e9i)T^{2} \) |
| 83 | \( 1 + (2.64e4 - 3.63e4i)T + (-1.21e9 - 3.74e9i)T^{2} \) |
| 89 | \( 1 + (1.57e4 + 4.85e4i)T + (-4.51e9 + 3.28e9i)T^{2} \) |
| 97 | \( 1 + (-4.12e4 - 5.67e4i)T + (-2.65e9 + 8.16e9i)T^{2} \) |
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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
−14.49787094129388718457291615574, −13.29304740911474741675986566536, −11.63444495436802175853409347965, −11.07806723096407467073824287175, −9.532927652630364707184804217081, −8.113300741888282865431053326186, −6.79238626120125618207121302418, −5.88392843022091998752757698091, −3.04766509210967152285103944030, −0.76275706592774619469504197865,
1.35601591589068831112088535986, 4.04510217496067042045750082187, 5.50753968404388703466972161764, 7.55029750395008776281581773523, 8.716726557700197811516987465143, 10.12561836463641251015286435179, 10.81441955540386939171223607872, 12.34577341735383576288673904687, 13.19714101268370357992138036066, 15.12051828816290121233671641448
