| L(s) = 1 | + (−0.149 + 1.71i)2-s + (1.03 + 0.182i)4-s + (2.66 − 4.23i)5-s + (−10.9 − 7.63i)7-s + (−2.24 + 8.37i)8-s + (6.84 + 5.18i)10-s + (5.04 + 1.83i)11-s + (−0.598 − 6.84i)13-s + (14.7 − 17.5i)14-s + (−10.0 − 3.66i)16-s + (−2.36 − 8.82i)17-s + (17.3 − 9.99i)19-s + (3.52 − 3.88i)20-s + (−3.90 + 8.36i)22-s + (8.21 − 5.75i)23-s + ⋯ |
| L(s) = 1 | + (−0.0748 + 0.855i)2-s + (0.258 + 0.0455i)4-s + (0.532 − 0.846i)5-s + (−1.55 − 1.09i)7-s + (−0.280 + 1.04i)8-s + (0.684 + 0.518i)10-s + (0.459 + 0.167i)11-s + (−0.0460 − 0.526i)13-s + (1.05 − 1.25i)14-s + (−0.628 − 0.228i)16-s + (−0.139 − 0.518i)17-s + (0.911 − 0.525i)19-s + (0.176 − 0.194i)20-s + (−0.177 + 0.380i)22-s + (0.357 − 0.250i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.769 + 0.638i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.769 + 0.638i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.39523 - 0.503028i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.39523 - 0.503028i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (-2.66 + 4.23i)T \) |
| good | 2 | \( 1 + (0.149 - 1.71i)T + (-3.93 - 0.694i)T^{2} \) |
| 7 | \( 1 + (10.9 + 7.63i)T + (16.7 + 46.0i)T^{2} \) |
| 11 | \( 1 + (-5.04 - 1.83i)T + (92.6 + 77.7i)T^{2} \) |
| 13 | \( 1 + (0.598 + 6.84i)T + (-166. + 29.3i)T^{2} \) |
| 17 | \( 1 + (2.36 + 8.82i)T + (-250. + 144.5i)T^{2} \) |
| 19 | \( 1 + (-17.3 + 9.99i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-8.21 + 5.75i)T + (180. - 497. i)T^{2} \) |
| 29 | \( 1 + (15.0 + 17.9i)T + (-146. + 828. i)T^{2} \) |
| 31 | \( 1 + (-9.98 + 56.6i)T + (-903. - 328. i)T^{2} \) |
| 37 | \( 1 + (12.0 + 44.9i)T + (-1.18e3 + 684.5i)T^{2} \) |
| 41 | \( 1 + (-15.9 - 13.3i)T + (291. + 1.65e3i)T^{2} \) |
| 43 | \( 1 + (-6.05 - 12.9i)T + (-1.18e3 + 1.41e3i)T^{2} \) |
| 47 | \( 1 + (-5.84 - 4.09i)T + (755. + 2.07e3i)T^{2} \) |
| 53 | \( 1 + (22.7 + 22.7i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + (-24.7 - 68.1i)T + (-2.66e3 + 2.23e3i)T^{2} \) |
| 61 | \( 1 + (3.42 + 19.4i)T + (-3.49e3 + 1.27e3i)T^{2} \) |
| 67 | \( 1 + (-37.5 + 3.28i)T + (4.42e3 - 779. i)T^{2} \) |
| 71 | \( 1 + (-23.5 + 40.7i)T + (-2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (19.9 - 74.2i)T + (-4.61e3 - 2.66e3i)T^{2} \) |
| 79 | \( 1 + (-25.2 - 30.1i)T + (-1.08e3 + 6.14e3i)T^{2} \) |
| 83 | \( 1 + (-32.6 - 2.85i)T + (6.78e3 + 1.19e3i)T^{2} \) |
| 89 | \( 1 + (111. - 64.5i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (52.8 - 24.6i)T + (6.04e3 - 7.20e3i)T^{2} \) |
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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
−10.85150972489483891332454679270, −9.669667725727006586174474096533, −9.285878450388020224374212064068, −7.913483715332154796387197087644, −7.10904380128770227310944299899, −6.29840471116052873912478393001, −5.44066301617783018030022726165, −4.07120914290620630413402870398, −2.64948352005774595177011703899, −0.63761834198674376871380399772,
1.71314850708270081769421355943, 2.96330038949387618972631738803, 3.48028188614636406834362985804, 5.61200740459433804833638519304, 6.48414857797823267412285976059, 7.01058529026278929820880189652, 8.855343614725198263598426105685, 9.601482258663942999866976723480, 10.18568976533187725949080295987, 11.12691355592291080673525736781
