| L(s) = 1 | + (1.71 + 5.26i)3-s + (5.02 + 3.65i)5-s + (−3.12 + 9.60i)7-s + (−2.98 + 2.17i)9-s + (−17.8 + 31.8i)11-s + (−16.0 + 11.6i)13-s + (−10.6 + 32.7i)15-s + (−0.0229 − 0.0166i)17-s + (22.3 + 68.8i)19-s − 55.9·21-s + 92.3·23-s + (−26.6 − 82.1i)25-s + (104. + 75.8i)27-s + (25.7 − 79.2i)29-s + (139. − 101. i)31-s + ⋯ |
| L(s) = 1 | + (0.329 + 1.01i)3-s + (0.449 + 0.326i)5-s + (−0.168 + 0.518i)7-s + (−0.110 + 0.0804i)9-s + (−0.489 + 0.871i)11-s + (−0.342 + 0.248i)13-s + (−0.183 + 0.563i)15-s + (−0.000327 − 0.000237i)17-s + (0.270 + 0.831i)19-s − 0.581·21-s + 0.837·23-s + (−0.213 − 0.656i)25-s + (0.744 + 0.540i)27-s + (0.164 − 0.507i)29-s + (0.806 − 0.585i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0924 - 0.995i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0924 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.13840 + 1.24895i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.13840 + 1.24895i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + (17.8 - 31.8i)T \) |
| good | 3 | \( 1 + (-1.71 - 5.26i)T + (-21.8 + 15.8i)T^{2} \) |
| 5 | \( 1 + (-5.02 - 3.65i)T + (38.6 + 118. i)T^{2} \) |
| 7 | \( 1 + (3.12 - 9.60i)T + (-277. - 201. i)T^{2} \) |
| 13 | \( 1 + (16.0 - 11.6i)T + (678. - 2.08e3i)T^{2} \) |
| 17 | \( 1 + (0.0229 + 0.0166i)T + (1.51e3 + 4.67e3i)T^{2} \) |
| 19 | \( 1 + (-22.3 - 68.8i)T + (-5.54e3 + 4.03e3i)T^{2} \) |
| 23 | \( 1 - 92.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + (-25.7 + 79.2i)T + (-1.97e4 - 1.43e4i)T^{2} \) |
| 31 | \( 1 + (-139. + 101. i)T + (9.20e3 - 2.83e4i)T^{2} \) |
| 37 | \( 1 + (-55.1 + 169. i)T + (-4.09e4 - 2.97e4i)T^{2} \) |
| 41 | \( 1 + (72.8 + 224. i)T + (-5.57e4 + 4.05e4i)T^{2} \) |
| 43 | \( 1 + 69.3T + 7.95e4T^{2} \) |
| 47 | \( 1 + (133. + 411. i)T + (-8.39e4 + 6.10e4i)T^{2} \) |
| 53 | \( 1 + (-485. + 352. i)T + (4.60e4 - 1.41e5i)T^{2} \) |
| 59 | \( 1 + (112. - 347. i)T + (-1.66e5 - 1.20e5i)T^{2} \) |
| 61 | \( 1 + (-554. - 402. i)T + (7.01e4 + 2.15e5i)T^{2} \) |
| 67 | \( 1 - 224.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (-206. - 149. i)T + (1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 + (272. - 839. i)T + (-3.14e5 - 2.28e5i)T^{2} \) |
| 79 | \( 1 + (313. - 227. i)T + (1.52e5 - 4.68e5i)T^{2} \) |
| 83 | \( 1 + (982. + 713. i)T + (1.76e5 + 5.43e5i)T^{2} \) |
| 89 | \( 1 + 1.23e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-721. + 524. i)T + (2.82e5 - 8.68e5i)T^{2} \) |
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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
−14.21285530035069622419042089344, −12.90766741748512983894250191087, −11.79118005145428862722249363959, −10.23203598063402823320284871510, −9.831853029299234126893791894975, −8.622217832936667002372562717030, −7.04911331594021165373326494413, −5.50119729924340288839096405579, −4.14217610226560311000014978732, −2.49434854920318690313435961363,
1.07287976651627275233622879383, 2.89244284341014989178280446935, 5.04355881376770893518048665774, 6.57742383281998157294935975027, 7.62846935234673218226195952226, 8.734000865637953971500064971707, 10.07611043391351073644625214451, 11.30141720585778367886427184685, 12.70584201183665267505144359039, 13.35824624598775136862036297993
