| L(s) = 1 | + (−4.11 + 2.99i)2-s + (−1.18 + 3.64i)3-s + (5.53 − 17.0i)4-s + (−10.3 − 4.17i)5-s + (−6.02 − 18.5i)6-s − 12.1·7-s + (15.5 + 47.9i)8-s + (9.98 + 7.25i)9-s + (55.1 − 13.8i)10-s + (−53.0 + 38.5i)11-s + (55.4 + 40.2i)12-s + (−24.5 − 17.8i)13-s + (50.1 − 36.4i)14-s + (27.4 − 32.8i)15-s + (−91.6 − 66.5i)16-s + (15.4 + 47.6i)17-s + ⋯ |
| L(s) = 1 | + (−1.45 + 1.05i)2-s + (−0.227 + 0.700i)3-s + (0.691 − 2.12i)4-s + (−0.927 − 0.373i)5-s + (−0.409 − 1.26i)6-s − 0.658·7-s + (0.688 + 2.11i)8-s + (0.369 + 0.268i)9-s + (1.74 − 0.436i)10-s + (−1.45 + 1.05i)11-s + (1.33 + 0.969i)12-s + (−0.523 − 0.380i)13-s + (0.958 − 0.696i)14-s + (0.473 − 0.564i)15-s + (−1.43 − 1.04i)16-s + (0.220 + 0.679i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 + 0.578i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.816 + 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0590560 - 0.185546i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0590560 - 0.185546i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (10.3 + 4.17i)T \) |
| good | 2 | \( 1 + (4.11 - 2.99i)T + (2.47 - 7.60i)T^{2} \) |
| 3 | \( 1 + (1.18 - 3.64i)T + (-21.8 - 15.8i)T^{2} \) |
| 7 | \( 1 + 12.1T + 343T^{2} \) |
| 11 | \( 1 + (53.0 - 38.5i)T + (411. - 1.26e3i)T^{2} \) |
| 13 | \( 1 + (24.5 + 17.8i)T + (678. + 2.08e3i)T^{2} \) |
| 17 | \( 1 + (-15.4 - 47.6i)T + (-3.97e3 + 2.88e3i)T^{2} \) |
| 19 | \( 1 + (-2.02 - 6.23i)T + (-5.54e3 + 4.03e3i)T^{2} \) |
| 23 | \( 1 + (-48.1 + 34.9i)T + (3.75e3 - 1.15e4i)T^{2} \) |
| 29 | \( 1 + (-48.3 + 148. i)T + (-1.97e4 - 1.43e4i)T^{2} \) |
| 31 | \( 1 + (-87.6 - 269. i)T + (-2.41e4 + 1.75e4i)T^{2} \) |
| 37 | \( 1 + (50.8 + 36.9i)T + (1.56e4 + 4.81e4i)T^{2} \) |
| 41 | \( 1 + (199. + 144. i)T + (2.12e4 + 6.55e4i)T^{2} \) |
| 43 | \( 1 + 38.7T + 7.95e4T^{2} \) |
| 47 | \( 1 + (44.4 - 136. i)T + (-8.39e4 - 6.10e4i)T^{2} \) |
| 53 | \( 1 + (-54.9 + 169. i)T + (-1.20e5 - 8.75e4i)T^{2} \) |
| 59 | \( 1 + (-124. - 90.5i)T + (6.34e4 + 1.95e5i)T^{2} \) |
| 61 | \( 1 + (99.3 - 72.1i)T + (7.01e4 - 2.15e5i)T^{2} \) |
| 67 | \( 1 + (-131. - 404. i)T + (-2.43e5 + 1.76e5i)T^{2} \) |
| 71 | \( 1 + (-79.0 + 243. i)T + (-2.89e5 - 2.10e5i)T^{2} \) |
| 73 | \( 1 + (-267. + 194. i)T + (1.20e5 - 3.69e5i)T^{2} \) |
| 79 | \( 1 + (225. - 695. i)T + (-3.98e5 - 2.89e5i)T^{2} \) |
| 83 | \( 1 + (137. + 422. i)T + (-4.62e5 + 3.36e5i)T^{2} \) |
| 89 | \( 1 + (838. - 609. i)T + (2.17e5 - 6.70e5i)T^{2} \) |
| 97 | \( 1 + (-33.5 + 103. i)T + (-7.38e5 - 5.36e5i)T^{2} \) |
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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
−17.58112731634023626346369519495, −16.45526735134962136712019423849, −15.66969581931589126401793510542, −15.14357697531218310239883743374, −12.66994115220324960108370041033, −10.56263289134674025562192533581, −9.853154043502882463058738730774, −8.239694388184092596439549975594, −7.16340468037931955919962685647, −5.04747092184341667329130129601,
0.31052054058991472183690078292, 3.02408896954651441976821244418, 7.08166980542807483213732475622, 8.154169791176923170366201831460, 9.793132937179102214791656626352, 11.07723642657205363543100767795, 12.06769336646069755896101837029, 13.17650188912344533059851217045, 15.67530300066406381332416677958, 16.69256973979527267849006477708
