| L(s) = 1 | + (−2.04 + 2.81i)2-s + (−8.11 + 2.63i)3-s + (−1.27 − 3.91i)4-s + (9.17 − 28.2i)6-s − 12.5i·7-s + (−12.8 − 4.17i)8-s + (37.0 − 26.9i)9-s + (−30.5 − 22.2i)11-s + (20.6 + 28.4i)12-s + (18.7 + 25.8i)13-s + (35.2 + 25.5i)14-s + (64.7 − 47.0i)16-s + (17.8 + 5.81i)17-s + 159. i·18-s + (−49.5 + 152. i)19-s + ⋯ |
| L(s) = 1 | + (−0.723 + 0.995i)2-s + (−1.56 + 0.507i)3-s + (−0.159 − 0.489i)4-s + (0.624 − 1.92i)6-s − 0.675i·7-s + (−0.567 − 0.184i)8-s + (1.37 − 0.996i)9-s + (−0.837 − 0.608i)11-s + (0.496 + 0.683i)12-s + (0.400 + 0.551i)13-s + (0.672 + 0.488i)14-s + (1.01 − 0.734i)16-s + (0.255 + 0.0829i)17-s + 2.08i·18-s + (−0.598 + 1.84i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.413782 + 0.131509i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.413782 + 0.131509i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 + (2.04 - 2.81i)T + (-2.47 - 7.60i)T^{2} \) |
| 3 | \( 1 + (8.11 - 2.63i)T + (21.8 - 15.8i)T^{2} \) |
| 7 | \( 1 + 12.5iT - 343T^{2} \) |
| 11 | \( 1 + (30.5 + 22.2i)T + (411. + 1.26e3i)T^{2} \) |
| 13 | \( 1 + (-18.7 - 25.8i)T + (-678. + 2.08e3i)T^{2} \) |
| 17 | \( 1 + (-17.8 - 5.81i)T + (3.97e3 + 2.88e3i)T^{2} \) |
| 19 | \( 1 + (49.5 - 152. i)T + (-5.54e3 - 4.03e3i)T^{2} \) |
| 23 | \( 1 + (-62.7 + 86.4i)T + (-3.75e3 - 1.15e4i)T^{2} \) |
| 29 | \( 1 + (33.1 + 102. i)T + (-1.97e4 + 1.43e4i)T^{2} \) |
| 31 | \( 1 + (2.18 - 6.73i)T + (-2.41e4 - 1.75e4i)T^{2} \) |
| 37 | \( 1 + (-38.0 - 52.3i)T + (-1.56e4 + 4.81e4i)T^{2} \) |
| 41 | \( 1 + (-305. + 222. i)T + (2.12e4 - 6.55e4i)T^{2} \) |
| 43 | \( 1 + 234. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (-101. + 32.9i)T + (8.39e4 - 6.10e4i)T^{2} \) |
| 53 | \( 1 + (-196. + 64.0i)T + (1.20e5 - 8.75e4i)T^{2} \) |
| 59 | \( 1 + (-261. + 190. i)T + (6.34e4 - 1.95e5i)T^{2} \) |
| 61 | \( 1 + (-221. - 161. i)T + (7.01e4 + 2.15e5i)T^{2} \) |
| 67 | \( 1 + (-134. - 43.7i)T + (2.43e5 + 1.76e5i)T^{2} \) |
| 71 | \( 1 + (185. + 570. i)T + (-2.89e5 + 2.10e5i)T^{2} \) |
| 73 | \( 1 + (84.9 - 116. i)T + (-1.20e5 - 3.69e5i)T^{2} \) |
| 79 | \( 1 + (98.8 + 304. i)T + (-3.98e5 + 2.89e5i)T^{2} \) |
| 83 | \( 1 + (1.05e3 + 344. i)T + (4.62e5 + 3.36e5i)T^{2} \) |
| 89 | \( 1 + (43.7 + 31.7i)T + (2.17e5 + 6.70e5i)T^{2} \) |
| 97 | \( 1 + (-1.25e3 + 406. i)T + (7.38e5 - 5.36e5i)T^{2} \) |
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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
−12.81375017244613446317168771762, −11.82022643922127264856725111179, −10.68185608588532492725172705478, −10.07226427173812455759905890557, −8.614963914213779002765314730632, −7.43978555620847908249537650195, −6.28808872919064344328004262993, −5.57657305320975779514657794613, −3.99869343718584488151524606937, −0.48772565296824917216425979050,
0.910056807299221988540075195240, 2.54752183404302133147518096338, 5.05404691706643599025482321469, 5.98721516053189270099044374169, 7.32098291150081473890591792078, 8.848289761994746307501704391485, 10.03116051421794196367889130836, 11.04001693050619887458387258810, 11.42645771232550301639455876700, 12.59428545340115939201971477652
