| L(s) = 1 | + (−1.68 − 1.08i)2-s + (−1.28 − 8.96i)3-s + (1.66 + 3.63i)4-s + (−3.51 − 1.03i)5-s + (−7.52 + 16.4i)6-s + (−6.62 + 7.64i)7-s + (1.13 − 7.91i)8-s + (−52.7 + 15.4i)9-s + (4.80 + 5.54i)10-s + (12.6 − 8.13i)11-s + (30.4 − 19.5i)12-s + (−40.8 − 47.1i)13-s + (19.4 − 5.70i)14-s + (−4.72 + 32.8i)15-s + (−10.4 + 12.0i)16-s + (43.2 − 94.7i)17-s + ⋯ |
| L(s) = 1 | + (−0.594 − 0.382i)2-s + (−0.248 − 1.72i)3-s + (0.207 + 0.454i)4-s + (−0.314 − 0.0924i)5-s + (−0.511 + 1.12i)6-s + (−0.357 + 0.412i)7-s + (0.0503 − 0.349i)8-s + (−1.95 + 0.573i)9-s + (0.151 + 0.175i)10-s + (0.346 − 0.222i)11-s + (0.733 − 0.471i)12-s + (−0.872 − 1.00i)13-s + (0.370 − 0.108i)14-s + (−0.0813 + 0.565i)15-s + (−0.163 + 0.188i)16-s + (0.617 − 1.35i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0310i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0310i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0101021 - 0.651349i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0101021 - 0.651349i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.68 + 1.08i)T \) |
| 23 | \( 1 + (-46.7 + 99.8i)T \) |
| good | 3 | \( 1 + (1.28 + 8.96i)T + (-25.9 + 7.60i)T^{2} \) |
| 5 | \( 1 + (3.51 + 1.03i)T + (105. + 67.5i)T^{2} \) |
| 7 | \( 1 + (6.62 - 7.64i)T + (-48.8 - 339. i)T^{2} \) |
| 11 | \( 1 + (-12.6 + 8.13i)T + (552. - 1.21e3i)T^{2} \) |
| 13 | \( 1 + (40.8 + 47.1i)T + (-312. + 2.17e3i)T^{2} \) |
| 17 | \( 1 + (-43.2 + 94.7i)T + (-3.21e3 - 3.71e3i)T^{2} \) |
| 19 | \( 1 + (-47.1 - 103. i)T + (-4.49e3 + 5.18e3i)T^{2} \) |
| 29 | \( 1 + (-1.48 + 3.25i)T + (-1.59e4 - 1.84e4i)T^{2} \) |
| 31 | \( 1 + (-46.0 + 320. i)T + (-2.85e4 - 8.39e3i)T^{2} \) |
| 37 | \( 1 + (95.1 - 27.9i)T + (4.26e4 - 2.73e4i)T^{2} \) |
| 41 | \( 1 + (-85.7 - 25.1i)T + (5.79e4 + 3.72e4i)T^{2} \) |
| 43 | \( 1 + (-62.0 - 431. i)T + (-7.62e4 + 2.23e4i)T^{2} \) |
| 47 | \( 1 - 67.1T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-383. + 442. i)T + (-2.11e4 - 1.47e5i)T^{2} \) |
| 59 | \( 1 + (-348. - 401. i)T + (-2.92e4 + 2.03e5i)T^{2} \) |
| 61 | \( 1 + (-7.25 + 50.4i)T + (-2.17e5 - 6.39e4i)T^{2} \) |
| 67 | \( 1 + (-334. - 214. i)T + (1.24e5 + 2.73e5i)T^{2} \) |
| 71 | \( 1 + (-31.9 - 20.5i)T + (1.48e5 + 3.25e5i)T^{2} \) |
| 73 | \( 1 + (-176. - 386. i)T + (-2.54e5 + 2.93e5i)T^{2} \) |
| 79 | \( 1 + (748. + 864. i)T + (-7.01e4 + 4.88e5i)T^{2} \) |
| 83 | \( 1 + (-243. + 71.4i)T + (4.81e5 - 3.09e5i)T^{2} \) |
| 89 | \( 1 + (99.9 + 695. i)T + (-6.76e5 + 1.98e5i)T^{2} \) |
| 97 | \( 1 + (-452. - 132. i)T + (7.67e5 + 4.93e5i)T^{2} \) |
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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
−14.47845683077803492604031680458, −13.14790650440627124272369104635, −12.20971733832495469359053758444, −11.61802639166822109748496451659, −9.802412640318732065718955702342, −8.151546772810980092544544754424, −7.35262070498880536221216617827, −5.87049835517993839972173326965, −2.65871821788950848810140671844, −0.63774516241290574917995054907,
3.77796239128478325811618245565, 5.23816319897105776361444575362, 7.04659177866944881402623224793, 8.913236699281579591078752532594, 9.789058285521653855303227527770, 10.73736082395905907087333781121, 11.88823312436473313618148284098, 14.07699100038639216880298377069, 15.15240180309384481001576238433, 15.83855664138571528545410310662
