| L(s) = 1 | + (7.08 + 5.15i)2-s + (5.03 + 15.5i)3-s + (13.8 + 42.5i)4-s + (5.66 − 55.6i)5-s + (−44.1 + 135. i)6-s − 72.9·7-s + (−34.6 + 106. i)8-s + (−18.4 + 13.3i)9-s + (326. − 365. i)10-s + (142. + 103. i)11-s + (−590. + 429. i)12-s + (4.89 − 3.55i)13-s + (−517. − 375. i)14-s + (890. − 192. i)15-s + (364. − 265. i)16-s + (−533. + 1.64e3i)17-s + ⋯ |
| L(s) = 1 | + (1.25 + 0.910i)2-s + (0.323 + 0.994i)3-s + (0.432 + 1.33i)4-s + (0.101 − 0.994i)5-s + (−0.500 + 1.54i)6-s − 0.562·7-s + (−0.191 + 0.588i)8-s + (−0.0758 + 0.0550i)9-s + (1.03 − 1.15i)10-s + (0.354 + 0.257i)11-s + (−1.18 + 0.860i)12-s + (0.00803 − 0.00584i)13-s + (−0.705 − 0.512i)14-s + (1.02 − 0.220i)15-s + (0.356 − 0.258i)16-s + (−0.447 + 1.37i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0145 - 0.999i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.0145 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.01773 + 1.98854i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.01773 + 1.98854i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-5.66 + 55.6i)T \) |
| good | 2 | \( 1 + (-7.08 - 5.15i)T + (9.88 + 30.4i)T^{2} \) |
| 3 | \( 1 + (-5.03 - 15.5i)T + (-196. + 142. i)T^{2} \) |
| 7 | \( 1 + 72.9T + 1.68e4T^{2} \) |
| 11 | \( 1 + (-142. - 103. i)T + (4.97e4 + 1.53e5i)T^{2} \) |
| 13 | \( 1 + (-4.89 + 3.55i)T + (1.14e5 - 3.53e5i)T^{2} \) |
| 17 | \( 1 + (533. - 1.64e3i)T + (-1.14e6 - 8.34e5i)T^{2} \) |
| 19 | \( 1 + (-753. + 2.31e3i)T + (-2.00e6 - 1.45e6i)T^{2} \) |
| 23 | \( 1 + (3.83e3 + 2.78e3i)T + (1.98e6 + 6.12e6i)T^{2} \) |
| 29 | \( 1 + (-388. - 1.19e3i)T + (-1.65e7 + 1.20e7i)T^{2} \) |
| 31 | \( 1 + (1.00e3 - 3.09e3i)T + (-2.31e7 - 1.68e7i)T^{2} \) |
| 37 | \( 1 + (1.24e4 - 9.02e3i)T + (2.14e7 - 6.59e7i)T^{2} \) |
| 41 | \( 1 + (4.21e3 - 3.06e3i)T + (3.58e7 - 1.10e8i)T^{2} \) |
| 43 | \( 1 - 1.46e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (2.27e3 + 6.99e3i)T + (-1.85e8 + 1.34e8i)T^{2} \) |
| 53 | \( 1 + (3.17e3 + 9.77e3i)T + (-3.38e8 + 2.45e8i)T^{2} \) |
| 59 | \( 1 + (-2.27e4 + 1.65e4i)T + (2.20e8 - 6.79e8i)T^{2} \) |
| 61 | \( 1 + (-1.65e4 - 1.20e4i)T + (2.60e8 + 8.03e8i)T^{2} \) |
| 67 | \( 1 + (8.77e3 - 2.70e4i)T + (-1.09e9 - 7.93e8i)T^{2} \) |
| 71 | \( 1 + (1.47e3 + 4.53e3i)T + (-1.45e9 + 1.06e9i)T^{2} \) |
| 73 | \( 1 + (-3.47e4 - 2.52e4i)T + (6.40e8 + 1.97e9i)T^{2} \) |
| 79 | \( 1 + (1.31e4 + 4.03e4i)T + (-2.48e9 + 1.80e9i)T^{2} \) |
| 83 | \( 1 + (-2.92e3 + 8.99e3i)T + (-3.18e9 - 2.31e9i)T^{2} \) |
| 89 | \( 1 + (-3.34e4 - 2.42e4i)T + (1.72e9 + 5.31e9i)T^{2} \) |
| 97 | \( 1 + (-4.31e4 - 1.32e5i)T + (-6.94e9 + 5.04e9i)T^{2} \) |
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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
−16.16812716966369974359902765194, −15.66506640474432703293318872506, −14.53793276987143955499129961727, −13.26124634703825498690141014090, −12.30136523314673098295909278201, −10.03299564650924174781368334055, −8.637222888090880947653924621398, −6.55016987461511667078275483698, −4.91037291874215251163252578733, −3.84009332315741245933022717603,
2.07961132493804352464384692818, 3.56342972713942639648401321104, 5.98561604857326384950132413379, 7.49090014580144272471366467290, 10.00057800541165232143588203349, 11.49544162803650017704837595167, 12.47381108154340959474195584285, 13.83922203148424883591568477483, 14.08698753600484126800399125443, 15.82897013204930675039603727162
