| L(s) = 1 | + (−4.96 + 0.876i)2-s + (−5.76 − 6.87i)3-s + (8.89 − 3.23i)4-s + (−10.5 − 3.82i)5-s + (34.6 + 29.0i)6-s + (28.6 − 49.6i)7-s + (28.5 − 16.4i)8-s + (0.0958 − 0.543i)9-s + (55.5 + 9.79i)10-s + (−29.1 − 50.5i)11-s + (−73.5 − 42.4i)12-s + (105. − 125. i)13-s + (−98.9 + 271. i)14-s + (34.2 + 94.2i)15-s + (−243. + 204. i)16-s + (31.2 + 176. i)17-s + ⋯ |
| L(s) = 1 | + (−1.24 + 0.219i)2-s + (−0.640 − 0.763i)3-s + (0.556 − 0.202i)4-s + (−0.420 − 0.152i)5-s + (0.963 + 0.808i)6-s + (0.585 − 1.01i)7-s + (0.446 − 0.257i)8-s + (0.00118 − 0.00670i)9-s + (0.555 + 0.0979i)10-s + (−0.241 − 0.417i)11-s + (−0.510 − 0.294i)12-s + (0.622 − 0.741i)13-s + (−0.505 + 1.38i)14-s + (0.152 + 0.418i)15-s + (−0.951 + 0.798i)16-s + (0.107 + 0.612i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.956 - 0.291i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.956 - 0.291i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.0438497 + 0.293844i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0438497 + 0.293844i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (10.5 + 3.82i)T \) |
| 19 | \( 1 + (240. + 269. i)T \) |
| good | 2 | \( 1 + (4.96 - 0.876i)T + (15.0 - 5.47i)T^{2} \) |
| 3 | \( 1 + (5.76 + 6.87i)T + (-14.0 + 79.7i)T^{2} \) |
| 7 | \( 1 + (-28.6 + 49.6i)T + (-1.20e3 - 2.07e3i)T^{2} \) |
| 11 | \( 1 + (29.1 + 50.5i)T + (-7.32e3 + 1.26e4i)T^{2} \) |
| 13 | \( 1 + (-105. + 125. i)T + (-4.95e3 - 2.81e4i)T^{2} \) |
| 17 | \( 1 + (-31.2 - 176. i)T + (-7.84e4 + 2.85e4i)T^{2} \) |
| 23 | \( 1 + (108. - 39.4i)T + (2.14e5 - 1.79e5i)T^{2} \) |
| 29 | \( 1 + (750. + 132. i)T + (6.64e5 + 2.41e5i)T^{2} \) |
| 31 | \( 1 + (723. + 417. i)T + (4.61e5 + 7.99e5i)T^{2} \) |
| 37 | \( 1 + 935. iT - 1.87e6T^{2} \) |
| 41 | \( 1 + (-1.70e3 - 2.02e3i)T + (-4.90e5 + 2.78e6i)T^{2} \) |
| 43 | \( 1 + (-2.24e3 - 815. i)T + (2.61e6 + 2.19e6i)T^{2} \) |
| 47 | \( 1 + (-3.41 + 19.3i)T + (-4.58e6 - 1.66e6i)T^{2} \) |
| 53 | \( 1 + (-117. - 322. i)T + (-6.04e6 + 5.07e6i)T^{2} \) |
| 59 | \( 1 + (3.37e3 - 595. i)T + (1.13e7 - 4.14e6i)T^{2} \) |
| 61 | \( 1 + (562. - 204. i)T + (1.06e7 - 8.89e6i)T^{2} \) |
| 67 | \( 1 + (7.75e3 + 1.36e3i)T + (1.89e7 + 6.89e6i)T^{2} \) |
| 71 | \( 1 + (2.36e3 - 6.49e3i)T + (-1.94e7 - 1.63e7i)T^{2} \) |
| 73 | \( 1 + (-6.09e3 + 5.11e3i)T + (4.93e6 - 2.79e7i)T^{2} \) |
| 79 | \( 1 + (2.35e3 + 2.80e3i)T + (-6.76e6 + 3.83e7i)T^{2} \) |
| 83 | \( 1 + (-2.61e3 + 4.53e3i)T + (-2.37e7 - 4.11e7i)T^{2} \) |
| 89 | \( 1 + (6.32e3 - 7.54e3i)T + (-1.08e7 - 6.17e7i)T^{2} \) |
| 97 | \( 1 + (1.42e4 - 2.51e3i)T + (8.31e7 - 3.02e7i)T^{2} \) |
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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
−12.72236628845281162586034814254, −11.14184066276548939574957565153, −10.72738746507099101550840576657, −9.192182082410491375372622670543, −7.961922275112947265864313463835, −7.37760871200002139133896600604, −6.04729634820253511525893074819, −4.12217672273724995132115936358, −1.25710754121317046171785276999, −0.26060783964941642893251279292,
1.96049594658021137388823321499, 4.37300298648747080091866629009, 5.60270995387759490513021201934, 7.46016020430585258189807887510, 8.593819514521363762838034369436, 9.486027529402978776431514626581, 10.65855635998425319541503883171, 11.24951124812215219675686896773, 12.26447242606389548904498597034, 14.00864964274443696526041237938
