| L(s) = 1 | + (0.972 + 0.495i)2-s + (0.872 + 0.138i)3-s + (−1.65 − 2.27i)4-s + (−2.66 + 4.23i)5-s + (0.779 + 0.566i)6-s + (1.62 − 1.62i)7-s + (−1.16 − 7.34i)8-s + (−7.81 − 2.54i)9-s + (−4.68 + 2.79i)10-s + (3.53 + 10.8i)11-s + (−1.12 − 2.20i)12-s + (7.63 − 3.89i)13-s + (2.39 − 0.776i)14-s + (−2.90 + 3.32i)15-s + (−0.963 + 2.96i)16-s + (25.1 − 3.98i)17-s + ⋯ |
| L(s) = 1 | + (0.486 + 0.247i)2-s + (0.290 + 0.0460i)3-s + (−0.412 − 0.567i)4-s + (−0.532 + 0.846i)5-s + (0.129 + 0.0944i)6-s + (0.232 − 0.232i)7-s + (−0.145 − 0.917i)8-s + (−0.868 − 0.282i)9-s + (−0.468 + 0.279i)10-s + (0.320 + 0.987i)11-s + (−0.0938 − 0.184i)12-s + (0.587 − 0.299i)13-s + (0.170 − 0.0554i)14-s + (−0.193 + 0.221i)15-s + (−0.0602 + 0.185i)16-s + (1.47 − 0.234i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.193i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.981 - 0.193i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.06696 + 0.104475i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.06696 + 0.104475i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (2.66 - 4.23i)T \) |
| good | 2 | \( 1 + (-0.972 - 0.495i)T + (2.35 + 3.23i)T^{2} \) |
| 3 | \( 1 + (-0.872 - 0.138i)T + (8.55 + 2.78i)T^{2} \) |
| 7 | \( 1 + (-1.62 + 1.62i)T - 49iT^{2} \) |
| 11 | \( 1 + (-3.53 - 10.8i)T + (-97.8 + 71.1i)T^{2} \) |
| 13 | \( 1 + (-7.63 + 3.89i)T + (99.3 - 136. i)T^{2} \) |
| 17 | \( 1 + (-25.1 + 3.98i)T + (274. - 89.3i)T^{2} \) |
| 19 | \( 1 + (5.60 - 7.71i)T + (-111. - 343. i)T^{2} \) |
| 23 | \( 1 + (5.39 - 10.5i)T + (-310. - 427. i)T^{2} \) |
| 29 | \( 1 + (-5.56 - 7.65i)T + (-259. + 799. i)T^{2} \) |
| 31 | \( 1 + (42.2 + 30.7i)T + (296. + 913. i)T^{2} \) |
| 37 | \( 1 + (21.6 + 42.4i)T + (-804. + 1.10e3i)T^{2} \) |
| 41 | \( 1 + (-16.6 + 51.2i)T + (-1.35e3 - 988. i)T^{2} \) |
| 43 | \( 1 + (-46.5 - 46.5i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (8.92 - 56.3i)T + (-2.10e3 - 682. i)T^{2} \) |
| 53 | \( 1 + (17.7 + 2.81i)T + (2.67e3 + 868. i)T^{2} \) |
| 59 | \( 1 + (13.2 + 4.30i)T + (2.81e3 + 2.04e3i)T^{2} \) |
| 61 | \( 1 + (-0.671 - 2.06i)T + (-3.01e3 + 2.18e3i)T^{2} \) |
| 67 | \( 1 + (-116. + 18.4i)T + (4.26e3 - 1.38e3i)T^{2} \) |
| 71 | \( 1 + (57.1 - 41.5i)T + (1.55e3 - 4.79e3i)T^{2} \) |
| 73 | \( 1 + (-28.5 + 56.0i)T + (-3.13e3 - 4.31e3i)T^{2} \) |
| 79 | \( 1 + (-19.9 - 27.4i)T + (-1.92e3 + 5.93e3i)T^{2} \) |
| 83 | \( 1 + (7.27 + 45.9i)T + (-6.55e3 + 2.12e3i)T^{2} \) |
| 89 | \( 1 + (30.3 - 9.84i)T + (6.40e3 - 4.65e3i)T^{2} \) |
| 97 | \( 1 + (-10.7 + 67.9i)T + (-8.94e3 - 2.90e3i)T^{2} \) |
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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
−17.56637785696073194779884322767, −15.79404460772429460552614651469, −14.55601403703903303004904591631, −14.27257112577544554237107433762, −12.45954206701400894343342456306, −10.89094010768245319584072984656, −9.480997137802061516929645504705, −7.59625204027150619256792919361, −5.85932041774539738289013968267, −3.78551683894993114844835411628,
3.54955360399321777443784806305, 5.36767013064634774912873572999, 8.162261812495775806862796431063, 8.824028590629797724861334441435, 11.31096096045243929638036204193, 12.28373900826838493943046107872, 13.55077643561507349917876369398, 14.52941563581730054658962644616, 16.33608268298649404076872121318, 17.09031050092095685258334964200
