| L(s) = 1 | + (−0.442 + 0.608i)2-s + (−6.75 + 2.19i)3-s + (2.29 + 7.07i)4-s + (1.00 + 11.1i)5-s + (1.65 − 5.08i)6-s − 18.3i·7-s + (−11.0 − 3.58i)8-s + (18.9 − 13.7i)9-s + (−7.22 − 4.31i)10-s + (34.1 + 24.8i)11-s + (−31.0 − 42.7i)12-s + (20.6 + 28.3i)13-s + (11.1 + 8.09i)14-s + (−31.2 − 73.0i)15-s + (−41.0 + 29.8i)16-s + (62.4 + 20.2i)17-s + ⋯ |
| L(s) = 1 | + (−0.156 + 0.215i)2-s + (−1.30 + 0.422i)3-s + (0.287 + 0.883i)4-s + (0.0898 + 0.995i)5-s + (0.112 − 0.345i)6-s − 0.988i·7-s + (−0.488 − 0.158i)8-s + (0.702 − 0.510i)9-s + (−0.228 − 0.136i)10-s + (0.936 + 0.680i)11-s + (−0.746 − 1.02i)12-s + (0.439 + 0.605i)13-s + (0.212 + 0.154i)14-s + (−0.537 − 1.25i)15-s + (−0.641 + 0.465i)16-s + (0.890 + 0.289i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.360 - 0.932i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.360 - 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.428251 + 0.624443i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.428251 + 0.624443i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-1.00 - 11.1i)T \) |
| good | 2 | \( 1 + (0.442 - 0.608i)T + (-2.47 - 7.60i)T^{2} \) |
| 3 | \( 1 + (6.75 - 2.19i)T + (21.8 - 15.8i)T^{2} \) |
| 7 | \( 1 + 18.3iT - 343T^{2} \) |
| 11 | \( 1 + (-34.1 - 24.8i)T + (411. + 1.26e3i)T^{2} \) |
| 13 | \( 1 + (-20.6 - 28.3i)T + (-678. + 2.08e3i)T^{2} \) |
| 17 | \( 1 + (-62.4 - 20.2i)T + (3.97e3 + 2.88e3i)T^{2} \) |
| 19 | \( 1 + (-29.2 + 90.1i)T + (-5.54e3 - 4.03e3i)T^{2} \) |
| 23 | \( 1 + (84.6 - 116. i)T + (-3.75e3 - 1.15e4i)T^{2} \) |
| 29 | \( 1 + (-28.1 - 86.7i)T + (-1.97e4 + 1.43e4i)T^{2} \) |
| 31 | \( 1 + (-99.7 + 306. i)T + (-2.41e4 - 1.75e4i)T^{2} \) |
| 37 | \( 1 + (-27.3 - 37.6i)T + (-1.56e4 + 4.81e4i)T^{2} \) |
| 41 | \( 1 + (68.8 - 50.0i)T + (2.12e4 - 6.55e4i)T^{2} \) |
| 43 | \( 1 - 84.2iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (-50.4 + 16.3i)T + (8.39e4 - 6.10e4i)T^{2} \) |
| 53 | \( 1 + (-552. + 179. i)T + (1.20e5 - 8.75e4i)T^{2} \) |
| 59 | \( 1 + (156. - 113. i)T + (6.34e4 - 1.95e5i)T^{2} \) |
| 61 | \( 1 + (297. + 216. i)T + (7.01e4 + 2.15e5i)T^{2} \) |
| 67 | \( 1 + (-277. - 90.2i)T + (2.43e5 + 1.76e5i)T^{2} \) |
| 71 | \( 1 + (239. + 738. i)T + (-2.89e5 + 2.10e5i)T^{2} \) |
| 73 | \( 1 + (-297. + 410. i)T + (-1.20e5 - 3.69e5i)T^{2} \) |
| 79 | \( 1 + (-383. - 1.17e3i)T + (-3.98e5 + 2.89e5i)T^{2} \) |
| 83 | \( 1 + (715. + 232. i)T + (4.62e5 + 3.36e5i)T^{2} \) |
| 89 | \( 1 + (204. + 148. i)T + (2.17e5 + 6.70e5i)T^{2} \) |
| 97 | \( 1 + (-354. + 115. i)T + (7.38e5 - 5.36e5i)T^{2} \) |
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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
−17.24879553160499704819077510805, −16.62652289217132469224095256081, −15.34416132945892946661547181853, −13.72735008240552066453114481430, −11.89806810283722354372787903393, −11.17862156380149921510965306273, −9.813960958742971392150717760353, −7.39554989946348339923815790120, −6.34740285607083139957709343927, −3.97176870124434260894441572332,
1.05239731101836253671545763143, 5.44099381243993746522562337936, 6.14821827215286521785481078129, 8.703611094807584114949433406779, 10.28045191105830443858219486643, 11.81713679341792778121436191563, 12.26819553387546818541382093289, 14.18509478889409353675516833980, 15.85032958657903232197597561557, 16.72668886078618427727290645647
