| L(s) = 1 | + (11.1 − 8.12i)2-s + (26.0 − 80.0i)3-s + (19.5 − 60.1i)4-s + (278. + 28.9i)5-s + (−359. − 1.10e3i)6-s + 483.·7-s + (276. + 851. i)8-s + (−3.96e3 − 2.87e3i)9-s + (3.34e3 − 1.93e3i)10-s + (−6.58e3 + 4.78e3i)11-s + (−4.30e3 − 3.13e3i)12-s + (2.59e3 + 1.88e3i)13-s + (5.40e3 − 3.92e3i)14-s + (9.54e3 − 2.15e4i)15-s + (1.65e4 + 1.20e4i)16-s + (2.55e3 + 7.85e3i)17-s + ⋯ |
| L(s) = 1 | + (0.988 − 0.718i)2-s + (0.556 − 1.71i)3-s + (0.152 − 0.470i)4-s + (0.994 + 0.103i)5-s + (−0.679 − 2.09i)6-s + 0.532·7-s + (0.191 + 0.587i)8-s + (−1.81 − 1.31i)9-s + (1.05 − 0.612i)10-s + (−1.49 + 1.08i)11-s + (−0.719 − 0.522i)12-s + (0.328 + 0.238i)13-s + (0.526 − 0.382i)14-s + (0.730 − 1.64i)15-s + (1.01 + 0.734i)16-s + (0.126 + 0.387i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.385 + 0.922i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.385 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.93675 - 2.90805i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.93675 - 2.90805i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-278. - 28.9i)T \) |
| good | 2 | \( 1 + (-11.1 + 8.12i)T + (39.5 - 121. i)T^{2} \) |
| 3 | \( 1 + (-26.0 + 80.0i)T + (-1.76e3 - 1.28e3i)T^{2} \) |
| 7 | \( 1 - 483.T + 8.23e5T^{2} \) |
| 11 | \( 1 + (6.58e3 - 4.78e3i)T + (6.02e6 - 1.85e7i)T^{2} \) |
| 13 | \( 1 + (-2.59e3 - 1.88e3i)T + (1.93e7 + 5.96e7i)T^{2} \) |
| 17 | \( 1 + (-2.55e3 - 7.85e3i)T + (-3.31e8 + 2.41e8i)T^{2} \) |
| 19 | \( 1 + (6.90e3 + 2.12e4i)T + (-7.23e8 + 5.25e8i)T^{2} \) |
| 23 | \( 1 + (-4.33e4 + 3.15e4i)T + (1.05e9 - 3.23e9i)T^{2} \) |
| 29 | \( 1 + (1.32e4 - 4.06e4i)T + (-1.39e10 - 1.01e10i)T^{2} \) |
| 31 | \( 1 + (-2.79e4 - 8.60e4i)T + (-2.22e10 + 1.61e10i)T^{2} \) |
| 37 | \( 1 + (4.31e5 + 3.13e5i)T + (2.93e10 + 9.02e10i)T^{2} \) |
| 41 | \( 1 + (4.38e5 + 3.18e5i)T + (6.01e10 + 1.85e11i)T^{2} \) |
| 43 | \( 1 - 5.34e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (9.60e4 - 2.95e5i)T + (-4.09e11 - 2.97e11i)T^{2} \) |
| 53 | \( 1 + (-3.43e5 + 1.05e6i)T + (-9.50e11 - 6.90e11i)T^{2} \) |
| 59 | \( 1 + (-1.59e6 - 1.15e6i)T + (7.69e11 + 2.36e12i)T^{2} \) |
| 61 | \( 1 + (1.13e6 - 8.24e5i)T + (9.71e11 - 2.98e12i)T^{2} \) |
| 67 | \( 1 + (-3.92e5 - 1.20e6i)T + (-4.90e12 + 3.56e12i)T^{2} \) |
| 71 | \( 1 + (-1.30e5 + 4.02e5i)T + (-7.35e12 - 5.34e12i)T^{2} \) |
| 73 | \( 1 + (-8.49e5 + 6.17e5i)T + (3.41e12 - 1.05e13i)T^{2} \) |
| 79 | \( 1 + (1.05e5 - 3.25e5i)T + (-1.55e13 - 1.12e13i)T^{2} \) |
| 83 | \( 1 + (-4.12e5 - 1.26e6i)T + (-2.19e13 + 1.59e13i)T^{2} \) |
| 89 | \( 1 + (6.34e6 - 4.61e6i)T + (1.36e13 - 4.20e13i)T^{2} \) |
| 97 | \( 1 + (1.53e6 - 4.71e6i)T + (-6.53e13 - 4.74e13i)T^{2} \) |
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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
−14.80826972765579002558978693719, −13.83166629409151782046008987338, −13.02929691033287765229110546870, −12.38989014918618314129310690243, −10.76077839548127191612211186434, −8.515134642473488352985376483311, −7.07372720613504188012743053730, −5.28632491328598456999036154776, −2.64017177469863223397235951887, −1.75226704301551730044148825535,
3.19470075592488207307485094352, 4.92112612072025939153394037084, 5.67535989879651741065278777819, 8.309019197789934259560986184465, 9.841920206958593477381091693653, 10.79793947708386892178097608783, 13.36044377033547416077246625883, 14.06509743384929492477472742803, 15.14796215506745334548874671938, 15.95970448251172733087717854042
