| L(s) = 1 | + (−3.31 + 0.887i)2-s + (−2.02 − 4.78i)3-s + (3.24 − 1.87i)4-s + (9.09 + 9.09i)5-s + (10.9 + 14.0i)6-s + (−7.29 + 27.2i)7-s + (10.3 − 10.3i)8-s + (−18.8 + 19.3i)9-s + (−38.2 − 22.0i)10-s + (8.81 + 32.8i)11-s + (−15.5 − 11.7i)12-s + (46.8 − 0.722i)13-s − 96.6i·14-s + (25.1 − 61.9i)15-s + (−39.9 + 69.2i)16-s + (−6.02 − 10.4i)17-s + ⋯ |
| L(s) = 1 | + (−1.17 + 0.313i)2-s + (−0.389 − 0.921i)3-s + (0.405 − 0.234i)4-s + (0.813 + 0.813i)5-s + (0.744 + 0.956i)6-s + (−0.393 + 1.47i)7-s + (0.455 − 0.455i)8-s + (−0.696 + 0.717i)9-s + (−1.20 − 0.697i)10-s + (0.241 + 0.901i)11-s + (−0.373 − 0.282i)12-s + (0.999 − 0.0154i)13-s − 1.84i·14-s + (0.432 − 1.06i)15-s + (−0.624 + 1.08i)16-s + (−0.0860 − 0.148i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.146 - 0.989i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.146 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.449110 + 0.387522i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.449110 + 0.387522i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.02 + 4.78i)T \) |
| 13 | \( 1 + (-46.8 + 0.722i)T \) |
| good | 2 | \( 1 + (3.31 - 0.887i)T + (6.92 - 4i)T^{2} \) |
| 5 | \( 1 + (-9.09 - 9.09i)T + 125iT^{2} \) |
| 7 | \( 1 + (7.29 - 27.2i)T + (-297. - 171.5i)T^{2} \) |
| 11 | \( 1 + (-8.81 - 32.8i)T + (-1.15e3 + 665.5i)T^{2} \) |
| 17 | \( 1 + (6.02 + 10.4i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (82.7 + 22.1i)T + (5.94e3 + 3.42e3i)T^{2} \) |
| 23 | \( 1 + (51.1 - 88.6i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-43.1 - 24.9i)T + (1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + (108. - 108. i)T - 2.97e4iT^{2} \) |
| 37 | \( 1 + (-369. + 98.9i)T + (4.38e4 - 2.53e4i)T^{2} \) |
| 41 | \( 1 + (94.7 - 25.3i)T + (5.96e4 - 3.44e4i)T^{2} \) |
| 43 | \( 1 + (-66.6 + 38.4i)T + (3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-222. + 222. i)T - 1.03e5iT^{2} \) |
| 53 | \( 1 + 276. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (-52.4 - 14.0i)T + (1.77e5 + 1.02e5i)T^{2} \) |
| 61 | \( 1 + (-156. - 270. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-26.8 - 100. i)T + (-2.60e5 + 1.50e5i)T^{2} \) |
| 71 | \( 1 + (-288. + 1.07e3i)T + (-3.09e5 - 1.78e5i)T^{2} \) |
| 73 | \( 1 + (-255. - 255. i)T + 3.89e5iT^{2} \) |
| 79 | \( 1 + 603.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (-616. - 616. i)T + 5.71e5iT^{2} \) |
| 89 | \( 1 + (-291. - 1.08e3i)T + (-6.10e5 + 3.52e5i)T^{2} \) |
| 97 | \( 1 + (1.03e3 + 278. i)T + (7.90e5 + 4.56e5i)T^{2} \) |
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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
−16.33102894703270439621546333039, −15.04749378400560145658089560954, −13.52322574822133346304644391500, −12.42837622747043705327125082602, −10.93572203538804785398163439629, −9.581735868275015825849629980515, −8.466610150702001026127799808703, −6.91088968052683638857250695749, −5.96631318003728336914666327018, −2.08830689056310696362862972867,
0.797805347622864858775111452066, 4.22471962252122372624222164599, 6.09502692766417097193215826150, 8.376274512954616610689406000828, 9.399247734036979889751861357881, 10.38591136061464078784409395680, 11.12957959464318335933383241166, 13.18077205260518673636601685070, 14.20697518552805071816545848256, 16.23511930443327553206414098283
