| L(s) = 1 | + (1.46 + 2.54i)2-s + (4.92 − 1.65i)3-s + (−0.305 + 0.529i)4-s − 3.69·5-s + (11.4 + 10.0i)6-s + (12.3 + 13.8i)7-s + 21.6·8-s + (21.4 − 16.3i)9-s + (−5.42 − 9.39i)10-s − 65.7·11-s + (−0.626 + 3.11i)12-s + (2.73 + 4.74i)13-s + (−16.9 + 51.6i)14-s + (−18.2 + 6.13i)15-s + (34.2 + 59.3i)16-s + (−25.1 − 43.5i)17-s + ⋯ |
| L(s) = 1 | + (0.518 + 0.898i)2-s + (0.947 − 0.319i)3-s + (−0.0382 + 0.0662i)4-s − 0.330·5-s + (0.778 + 0.685i)6-s + (0.666 + 0.745i)7-s + 0.958·8-s + (0.795 − 0.605i)9-s + (−0.171 − 0.297i)10-s − 1.80·11-s + (−0.0150 + 0.0749i)12-s + (0.0584 + 0.101i)13-s + (−0.323 + 0.985i)14-s + (−0.313 + 0.105i)15-s + (0.535 + 0.927i)16-s + (−0.358 − 0.621i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.734 - 0.678i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.734 - 0.678i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.26767 + 0.886718i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.26767 + 0.886718i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-4.92 + 1.65i)T \) |
| 7 | \( 1 + (-12.3 - 13.8i)T \) |
| good | 2 | \( 1 + (-1.46 - 2.54i)T + (-4 + 6.92i)T^{2} \) |
| 5 | \( 1 + 3.69T + 125T^{2} \) |
| 11 | \( 1 + 65.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + (-2.73 - 4.74i)T + (-1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + (25.1 + 43.5i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-0.769 + 1.33i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + 120.T + 1.21e4T^{2} \) |
| 29 | \( 1 + (39.2 - 68.0i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-151. + 262. i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-96.6 + 167. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-196. - 340. i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (138. - 239. i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-126. - 218. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (204. + 353. i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (131. - 227. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-56.1 - 97.2i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-49.1 + 85.1i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + 255.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-344. - 596. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-542. - 939. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-152. + 263. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + (-550. + 953. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-493. + 855. i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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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
−14.71420982520893917283485327089, −13.72390897789620005588522746552, −12.85184272234082054511078100347, −11.32181514949756554525858695396, −9.801527296997346846838469640388, −8.106976314787953499433413923169, −7.63920251459518517537098297153, −5.95342810291360295633967784332, −4.56296557571688192515591452369, −2.35016298971812639031798808618,
2.17044391967789407382330913712, 3.66149240047020816178098867069, 4.81169388978896630496223351169, 7.58069912438241956970298278359, 8.204196922902559581943761617982, 10.24151144112092458628701134641, 10.74848983604842003673236597846, 12.19961647811642387626897047344, 13.38303284149352343028680101565, 13.90354170690395565349774179989
