| L(s) = 1 | + (1.19 + 0.752i)2-s + (4.47 − 1.56i)3-s + (0.867 + 1.80i)4-s + (−4.81 + 1.09i)5-s + (6.53 + 1.49i)6-s + (−11.2 − 5.39i)7-s + (−0.316 + 2.81i)8-s + (10.5 − 8.41i)9-s + (−6.59 − 2.30i)10-s + (1.51 + 13.4i)11-s + (6.70 + 6.70i)12-s + (6.16 + 4.91i)13-s + (−9.35 − 14.8i)14-s + (−19.8 + 12.4i)15-s + (−2.49 + 3.12i)16-s + (14.2 − 14.2i)17-s + ⋯ |
| L(s) = 1 | + (0.598 + 0.376i)2-s + (1.49 − 0.522i)3-s + (0.216 + 0.450i)4-s + (−0.962 + 0.219i)5-s + (1.08 + 0.248i)6-s + (−1.60 − 0.770i)7-s + (−0.0395 + 0.351i)8-s + (1.17 − 0.935i)9-s + (−0.659 − 0.230i)10-s + (0.137 + 1.22i)11-s + (0.558 + 0.558i)12-s + (0.474 + 0.378i)13-s + (−0.668 − 1.06i)14-s + (−1.32 + 0.830i)15-s + (−0.155 + 0.195i)16-s + (0.839 − 0.839i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.177i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.984 - 0.177i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.87966 + 0.168081i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.87966 + 0.168081i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.19 - 0.752i)T \) |
| 29 | \( 1 + (-11.6 - 26.5i)T \) |
| good | 3 | \( 1 + (-4.47 + 1.56i)T + (7.03 - 5.61i)T^{2} \) |
| 5 | \( 1 + (4.81 - 1.09i)T + (22.5 - 10.8i)T^{2} \) |
| 7 | \( 1 + (11.2 + 5.39i)T + (30.5 + 38.3i)T^{2} \) |
| 11 | \( 1 + (-1.51 - 13.4i)T + (-117. + 26.9i)T^{2} \) |
| 13 | \( 1 + (-6.16 - 4.91i)T + (37.6 + 164. i)T^{2} \) |
| 17 | \( 1 + (-14.2 + 14.2i)T - 289iT^{2} \) |
| 19 | \( 1 + (-0.424 + 1.21i)T + (-282. - 225. i)T^{2} \) |
| 23 | \( 1 + (-7.52 + 32.9i)T + (-476. - 229. i)T^{2} \) |
| 31 | \( 1 + (7.54 + 4.74i)T + (416. + 865. i)T^{2} \) |
| 37 | \( 1 + (1.41 - 12.5i)T + (-1.33e3 - 304. i)T^{2} \) |
| 41 | \( 1 + (14.0 + 14.0i)T + 1.68e3iT^{2} \) |
| 43 | \( 1 + (30.1 + 48.0i)T + (-802. + 1.66e3i)T^{2} \) |
| 47 | \( 1 + (-21.8 + 2.46i)T + (2.15e3 - 491. i)T^{2} \) |
| 53 | \( 1 + (-4.69 - 20.5i)T + (-2.53e3 + 1.21e3i)T^{2} \) |
| 59 | \( 1 - 93.2T + 3.48e3T^{2} \) |
| 61 | \( 1 + (56.0 - 19.6i)T + (2.90e3 - 2.32e3i)T^{2} \) |
| 67 | \( 1 + (78.6 - 62.7i)T + (998. - 4.37e3i)T^{2} \) |
| 71 | \( 1 + (-12.2 - 9.76i)T + (1.12e3 + 4.91e3i)T^{2} \) |
| 73 | \( 1 + (40.3 - 25.3i)T + (2.31e3 - 4.80e3i)T^{2} \) |
| 79 | \( 1 + (27.9 + 3.14i)T + (6.08e3 + 1.38e3i)T^{2} \) |
| 83 | \( 1 + (14.3 - 6.91i)T + (4.29e3 - 5.38e3i)T^{2} \) |
| 89 | \( 1 + (37.4 + 23.5i)T + (3.43e3 + 7.13e3i)T^{2} \) |
| 97 | \( 1 + (-26.5 - 9.29i)T + (7.35e3 + 5.86e3i)T^{2} \) |
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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
−14.81768555618461551006662740644, −13.89683762888685956918595604564, −12.97032862534875575226734346070, −12.12832536457800291023832464342, −10.10252952238234345807893913389, −8.803826610853398547107612594840, −7.34340528050805853883632368578, −6.90323454169123513837999831038, −4.04741426964797369487637225565, −3.02081191714816438376127337458,
3.16531759327642514349517823740, 3.68133584933894487813791838021, 5.96177668667451057757847175715, 7.990138455893654726106872116760, 9.029185601310244809009719345470, 10.08092417004416618240162444872, 11.68529916438958032855335283529, 12.91330008809697231164545094233, 13.69819980832765207961870692246, 15.00357656964582877791498585374
