| L(s) = 1 | + (−3.80 + 1.23i)2-s + (4.87 + 6.70i)3-s + (12.9 − 9.40i)4-s + (44.0 + 34.4i)5-s + (−26.8 − 19.4i)6-s − 63.1i·7-s + (−37.6 + 51.7i)8-s + (53.8 − 165. i)9-s + (−210. − 76.7i)10-s + (180. + 555. i)11-s + (126. + 40.9i)12-s + (358. + 116. i)13-s + (78.1 + 240. i)14-s + (−16.8 + 463. i)15-s + (79.1 − 243. i)16-s + (−1.32e3 + 1.82e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.672 + 0.218i)2-s + (0.312 + 0.430i)3-s + (0.404 − 0.293i)4-s + (0.787 + 0.616i)5-s + (−0.304 − 0.221i)6-s − 0.487i·7-s + (−0.207 + 0.286i)8-s + (0.221 − 0.681i)9-s + (−0.664 − 0.242i)10-s + (0.449 + 1.38i)11-s + (0.252 + 0.0821i)12-s + (0.587 + 0.190i)13-s + (0.106 + 0.327i)14-s + (−0.0193 + 0.531i)15-s + (0.0772 − 0.237i)16-s + (−1.11 + 1.53i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.318 - 0.947i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.318 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.25625 + 0.903387i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.25625 + 0.903387i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (3.80 - 1.23i)T \) |
| 5 | \( 1 + (-44.0 - 34.4i)T \) |
| good | 3 | \( 1 + (-4.87 - 6.70i)T + (-75.0 + 231. i)T^{2} \) |
| 7 | \( 1 + 63.1iT - 1.68e4T^{2} \) |
| 11 | \( 1 + (-180. - 555. i)T + (-1.30e5 + 9.46e4i)T^{2} \) |
| 13 | \( 1 + (-358. - 116. i)T + (3.00e5 + 2.18e5i)T^{2} \) |
| 17 | \( 1 + (1.32e3 - 1.82e3i)T + (-4.38e5 - 1.35e6i)T^{2} \) |
| 19 | \( 1 + (-1.68e3 - 1.22e3i)T + (7.65e5 + 2.35e6i)T^{2} \) |
| 23 | \( 1 + (118. - 38.4i)T + (5.20e6 - 3.78e6i)T^{2} \) |
| 29 | \( 1 + (-3.33e3 + 2.42e3i)T + (6.33e6 - 1.95e7i)T^{2} \) |
| 31 | \( 1 + (5.27e3 + 3.83e3i)T + (8.84e6 + 2.72e7i)T^{2} \) |
| 37 | \( 1 + (1.04e4 + 3.38e3i)T + (5.61e7 + 4.07e7i)T^{2} \) |
| 41 | \( 1 + (-3.83e3 + 1.18e4i)T + (-9.37e7 - 6.80e7i)T^{2} \) |
| 43 | \( 1 + 1.57e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + (-1.52e4 - 2.09e4i)T + (-7.08e7 + 2.18e8i)T^{2} \) |
| 53 | \( 1 + (1.52e3 + 2.10e3i)T + (-1.29e8 + 3.97e8i)T^{2} \) |
| 59 | \( 1 + (-4.13e3 + 1.27e4i)T + (-5.78e8 - 4.20e8i)T^{2} \) |
| 61 | \( 1 + (9.74e3 + 3.00e4i)T + (-6.83e8 + 4.96e8i)T^{2} \) |
| 67 | \( 1 + (-1.09e4 + 1.50e4i)T + (-4.17e8 - 1.28e9i)T^{2} \) |
| 71 | \( 1 + (-1.40e4 + 1.02e4i)T + (5.57e8 - 1.71e9i)T^{2} \) |
| 73 | \( 1 + (7.21e4 - 2.34e4i)T + (1.67e9 - 1.21e9i)T^{2} \) |
| 79 | \( 1 + (-3.58e4 + 2.60e4i)T + (9.50e8 - 2.92e9i)T^{2} \) |
| 83 | \( 1 + (-4.03e4 + 5.55e4i)T + (-1.21e9 - 3.74e9i)T^{2} \) |
| 89 | \( 1 + (1.91e4 + 5.90e4i)T + (-4.51e9 + 3.28e9i)T^{2} \) |
| 97 | \( 1 + (9.82e3 + 1.35e4i)T + (-2.65e9 + 8.16e9i)T^{2} \) |
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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
−14.90691411367959023277332785901, −13.96289535978926038286532847722, −12.43551902355549210767331102432, −10.77511501252144388947267387794, −9.901566968894986246935666491565, −8.998018737608502433945801418004, −7.25566035017761397101828920514, −6.16583488833701044677515046874, −3.92554555914541587762793454667, −1.75882987322707852090943476622,
1.07144222919609917657023906989, 2.70831621955414935117922738571, 5.30581044116127427167984524332, 6.91649724423436592594047849732, 8.566162461267863126338626285595, 9.150095441045965555954762307302, 10.73311113019922986617061089793, 11.89336234441057553711182846030, 13.39030688246491952524397528880, 13.85766018797754287853173726895
