| L(s) = 1 | + (−1.46 + 1.36i)2-s + (−2.91 + 0.439i)3-s + (0.298 − 3.98i)4-s + (−5.72 + 14.5i)5-s + (3.67 − 4.61i)6-s + (18.2 − 3.41i)7-s + (4.98 + 6.25i)8-s + (−17.4 + 5.39i)9-s + (−11.4 − 29.1i)10-s + (−32.6 − 10.0i)11-s + (0.881 + 11.7i)12-s + (−20.2 − 88.5i)13-s + (−22.0 + 29.7i)14-s + (10.2 − 45.0i)15-s + (−15.8 − 2.38i)16-s + (−20.0 + 13.6i)17-s + ⋯ |
| L(s) = 1 | + (−0.518 + 0.480i)2-s + (−0.561 + 0.0845i)3-s + (0.0373 − 0.498i)4-s + (−0.511 + 1.30i)5-s + (0.250 − 0.313i)6-s + (0.982 − 0.184i)7-s + (0.220 + 0.276i)8-s + (−0.647 + 0.199i)9-s + (−0.361 − 0.922i)10-s + (−0.895 − 0.276i)11-s + (0.0212 + 0.282i)12-s + (−0.431 − 1.88i)13-s + (−0.420 + 0.568i)14-s + (0.176 − 0.775i)15-s + (−0.247 − 0.0372i)16-s + (−0.286 + 0.195i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.530 + 0.847i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.530 + 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0150088 - 0.0270889i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0150088 - 0.0270889i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.46 - 1.36i)T \) |
| 7 | \( 1 + (-18.2 + 3.41i)T \) |
| good | 3 | \( 1 + (2.91 - 0.439i)T + (25.8 - 7.95i)T^{2} \) |
| 5 | \( 1 + (5.72 - 14.5i)T + (-91.6 - 85.0i)T^{2} \) |
| 11 | \( 1 + (32.6 + 10.0i)T + (1.09e3 + 749. i)T^{2} \) |
| 13 | \( 1 + (20.2 + 88.5i)T + (-1.97e3 + 953. i)T^{2} \) |
| 17 | \( 1 + (20.0 - 13.6i)T + (1.79e3 - 4.57e3i)T^{2} \) |
| 19 | \( 1 + (8.71 + 15.0i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (44.1 + 30.1i)T + (4.44e3 + 1.13e4i)T^{2} \) |
| 29 | \( 1 + (182. + 87.8i)T + (1.52e4 + 1.90e4i)T^{2} \) |
| 31 | \( 1 + (-6.37 + 11.0i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-11.1 - 148. i)T + (-5.00e4 + 7.54e3i)T^{2} \) |
| 41 | \( 1 + (-147. - 185. i)T + (-1.53e4 + 6.71e4i)T^{2} \) |
| 43 | \( 1 + (271. - 340. i)T + (-1.76e4 - 7.75e4i)T^{2} \) |
| 47 | \( 1 + (293. - 271. i)T + (7.75e3 - 1.03e5i)T^{2} \) |
| 53 | \( 1 + (-13.3 + 178. i)T + (-1.47e5 - 2.21e4i)T^{2} \) |
| 59 | \( 1 + (104. + 267. i)T + (-1.50e5 + 1.39e5i)T^{2} \) |
| 61 | \( 1 + (38.4 + 513. i)T + (-2.24e5 + 3.38e4i)T^{2} \) |
| 67 | \( 1 + (239. - 414. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + (861. - 415. i)T + (2.23e5 - 2.79e5i)T^{2} \) |
| 73 | \( 1 + (517. + 480. i)T + (2.90e4 + 3.87e5i)T^{2} \) |
| 79 | \( 1 + (-510. - 884. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-226. + 992. i)T + (-5.15e5 - 2.48e5i)T^{2} \) |
| 89 | \( 1 + (-615. + 189. i)T + (5.82e5 - 3.97e5i)T^{2} \) |
| 97 | \( 1 + 1.22e3T + 9.12e5T^{2} \) |
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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
−13.18478989252598831146082226567, −11.49430392666892533296847436692, −10.88484619139446026971050707546, −10.17920866499487087364564414202, −8.123232302477475852667354493156, −7.70175874042166286462155099123, −6.17054359410877972163039950569, −5.05057530478551558877992782840, −2.84926785182054555488959294419, −0.02157330618823600316309211209,
1.83929621290948143886384651870, 4.33393278256901827471774049229, 5.36859156532955520820555835300, 7.31600369542566017425557044269, 8.528808462228778740539044147059, 9.216306401681672524744802630517, 10.84475825098968369420717501799, 11.82799715388588829010148638370, 12.19180288467243425292588970888, 13.55143229000986399289412244552
