| L(s) = 1 | + (4.41 + 0.386i)2-s + (4.70 − 2.21i)3-s + (11.4 + 2.02i)4-s + (−1.20 + 11.1i)5-s + (21.6 − 7.95i)6-s + (4.39 + 3.07i)7-s + (15.7 + 4.21i)8-s + (17.2 − 20.7i)9-s + (−9.63 + 48.6i)10-s + (−3.60 + 9.90i)11-s + (58.5 − 15.8i)12-s + (−3.16 − 36.1i)13-s + (18.2 + 15.2i)14-s + (18.8 + 54.9i)15-s + (−19.9 − 7.24i)16-s + (−82.8 + 22.1i)17-s + ⋯ |
| L(s) = 1 | + (1.56 + 0.136i)2-s + (0.904 − 0.425i)3-s + (1.43 + 0.253i)4-s + (−0.108 + 0.994i)5-s + (1.47 − 0.541i)6-s + (0.237 + 0.166i)7-s + (0.694 + 0.186i)8-s + (0.637 − 0.770i)9-s + (−0.304 + 1.53i)10-s + (−0.0988 + 0.271i)11-s + (1.40 − 0.382i)12-s + (−0.0675 − 0.771i)13-s + (0.347 + 0.291i)14-s + (0.325 + 0.945i)15-s + (−0.311 − 0.113i)16-s + (−1.18 + 0.316i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.190i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.981 - 0.190i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.63963 + 0.445793i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.63963 + 0.445793i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-4.70 + 2.21i)T \) |
| 5 | \( 1 + (1.20 - 11.1i)T \) |
| good | 2 | \( 1 + (-4.41 - 0.386i)T + (7.87 + 1.38i)T^{2} \) |
| 7 | \( 1 + (-4.39 - 3.07i)T + (117. + 322. i)T^{2} \) |
| 11 | \( 1 + (3.60 - 9.90i)T + (-1.01e3 - 855. i)T^{2} \) |
| 13 | \( 1 + (3.16 + 36.1i)T + (-2.16e3 + 381. i)T^{2} \) |
| 17 | \( 1 + (82.8 - 22.1i)T + (4.25e3 - 2.45e3i)T^{2} \) |
| 19 | \( 1 + (-76.5 + 44.2i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (39.3 + 56.1i)T + (-4.16e3 + 1.14e4i)T^{2} \) |
| 29 | \( 1 + (-41.1 + 34.5i)T + (4.23e3 - 2.40e4i)T^{2} \) |
| 31 | \( 1 + (4.82 - 27.3i)T + (-2.79e4 - 1.01e4i)T^{2} \) |
| 37 | \( 1 + (-54.1 - 201. i)T + (-4.38e4 + 2.53e4i)T^{2} \) |
| 41 | \( 1 + (207. - 247. i)T + (-1.19e4 - 6.78e4i)T^{2} \) |
| 43 | \( 1 + (-189. - 405. i)T + (-5.11e4 + 6.09e4i)T^{2} \) |
| 47 | \( 1 + (20.4 - 29.2i)T + (-3.55e4 - 9.75e4i)T^{2} \) |
| 53 | \( 1 + (-150. + 150. i)T - 1.48e5iT^{2} \) |
| 59 | \( 1 + (211. - 76.9i)T + (1.57e5 - 1.32e5i)T^{2} \) |
| 61 | \( 1 + (-61.4 - 348. i)T + (-2.13e5 + 7.76e4i)T^{2} \) |
| 67 | \( 1 + (-463. + 40.5i)T + (2.96e5 - 5.22e4i)T^{2} \) |
| 71 | \( 1 + (-659. - 380. i)T + (1.78e5 + 3.09e5i)T^{2} \) |
| 73 | \( 1 + (-303. + 1.13e3i)T + (-3.36e5 - 1.94e5i)T^{2} \) |
| 79 | \( 1 + (829. + 988. i)T + (-8.56e4 + 4.85e5i)T^{2} \) |
| 83 | \( 1 + (110. - 1.25e3i)T + (-5.63e5 - 9.92e4i)T^{2} \) |
| 89 | \( 1 + (-315. - 546. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (971. - 453. i)T + (5.86e5 - 6.99e5i)T^{2} \) |
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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
−13.15506601461063402220727386852, −12.10858027774679463969566618257, −11.13473908742056698026244197383, −9.758721192084828165246952732684, −8.206992033309534730147116265911, −7.06819960693867165803342350309, −6.22682590368660015661094516052, −4.63962433317135299884061036714, −3.32181868158032935565817330311, −2.40872341965310956460434149629,
2.06441802090624286852667073828, 3.68278925781679311302770093968, 4.52184557864810612227239845445, 5.52181883607366494747557503995, 7.20643043102173357831127109378, 8.575396937715010205039915806067, 9.494821062971905504755331425517, 11.03449974926007324847362606593, 12.03373424811374163793643613803, 12.98628683100328355886942561589
