| L(s) = 1 | + (1.48 + 4.58i)2-s + (−2.42 + 1.76i)3-s + (−12.3 + 8.93i)4-s + (5.89 + 9.49i)5-s + (−11.6 − 8.49i)6-s + 1.13·7-s + (−28.0 − 20.4i)8-s + (2.78 − 8.55i)9-s + (−34.7 + 41.1i)10-s + (−17.7 − 54.7i)11-s + (14.0 − 43.3i)12-s + (−12.9 + 39.9i)13-s + (1.68 + 5.19i)14-s + (−31.0 − 12.6i)15-s + (14.0 − 43.3i)16-s + (87.1 + 63.3i)17-s + ⋯ |
| L(s) = 1 | + (0.526 + 1.61i)2-s + (−0.467 + 0.339i)3-s + (−1.53 + 1.11i)4-s + (0.527 + 0.849i)5-s + (−0.795 − 0.577i)6-s + 0.0612·7-s + (−1.24 − 0.901i)8-s + (0.103 − 0.317i)9-s + (−1.09 + 1.30i)10-s + (−0.487 − 1.49i)11-s + (0.339 − 1.04i)12-s + (−0.276 + 0.851i)13-s + (0.0322 + 0.0992i)14-s + (−0.534 − 0.217i)15-s + (0.219 − 0.677i)16-s + (1.24 + 0.903i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 + 0.229i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.179355 - 1.53975i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.179355 - 1.53975i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.42 - 1.76i)T \) |
| 5 | \( 1 + (-5.89 - 9.49i)T \) |
| good | 2 | \( 1 + (-1.48 - 4.58i)T + (-6.47 + 4.70i)T^{2} \) |
| 7 | \( 1 - 1.13T + 343T^{2} \) |
| 11 | \( 1 + (17.7 + 54.7i)T + (-1.07e3 + 782. i)T^{2} \) |
| 13 | \( 1 + (12.9 - 39.9i)T + (-1.77e3 - 1.29e3i)T^{2} \) |
| 17 | \( 1 + (-87.1 - 63.3i)T + (1.51e3 + 4.67e3i)T^{2} \) |
| 19 | \( 1 + (-43.1 - 31.3i)T + (2.11e3 + 6.52e3i)T^{2} \) |
| 23 | \( 1 + (5.32 + 16.3i)T + (-9.84e3 + 7.15e3i)T^{2} \) |
| 29 | \( 1 + (113. - 82.4i)T + (7.53e3 - 2.31e4i)T^{2} \) |
| 31 | \( 1 + (-196. - 142. i)T + (9.20e3 + 2.83e4i)T^{2} \) |
| 37 | \( 1 + (-107. + 332. i)T + (-4.09e4 - 2.97e4i)T^{2} \) |
| 41 | \( 1 + (2.64 - 8.13i)T + (-5.57e4 - 4.05e4i)T^{2} \) |
| 43 | \( 1 - 111.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-261. + 189. i)T + (3.20e4 - 9.87e4i)T^{2} \) |
| 53 | \( 1 + (-332. + 241. i)T + (4.60e4 - 1.41e5i)T^{2} \) |
| 59 | \( 1 + (-205. + 633. i)T + (-1.66e5 - 1.20e5i)T^{2} \) |
| 61 | \( 1 + (-91.5 - 281. i)T + (-1.83e5 + 1.33e5i)T^{2} \) |
| 67 | \( 1 + (-679. - 494. i)T + (9.29e4 + 2.86e5i)T^{2} \) |
| 71 | \( 1 + (-45.5 + 33.1i)T + (1.10e5 - 3.40e5i)T^{2} \) |
| 73 | \( 1 + (-66.9 - 206. i)T + (-3.14e5 + 2.28e5i)T^{2} \) |
| 79 | \( 1 + (339. - 246. i)T + (1.52e5 - 4.68e5i)T^{2} \) |
| 83 | \( 1 + (1.17e3 + 856. i)T + (1.76e5 + 5.43e5i)T^{2} \) |
| 89 | \( 1 + (378. + 1.16e3i)T + (-5.70e5 + 4.14e5i)T^{2} \) |
| 97 | \( 1 + (419. - 305. i)T + (2.82e5 - 8.68e5i)T^{2} \) |
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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
−14.42850713233980569012489670390, −14.17370274212873097294155281131, −12.88444562241489238268569046040, −11.31117366872347776178765429658, −10.07760609048031085524937676264, −8.571221722930800381852756109086, −7.29313505127918414608440966931, −6.10301228041794696139038231758, −5.42537191144666905194970306427, −3.62502122129570151676902701033,
0.972524553313580678037349860698, 2.53087620853950866811767077835, 4.61990255949662002582055271446, 5.47431362272067872870435784165, 7.67857350567864834469552822208, 9.652220269285908219581117312065, 10.08168110732042748155878282427, 11.59721803108930207091382825544, 12.35609058701998971271751649712, 13.04061400027961912628406842194
