| L(s) = 1 | + (0.309 + 0.951i)2-s + (0.689 − 0.500i)3-s + (−0.809 + 0.587i)4-s + (−1.78 + 1.34i)5-s + (0.689 + 0.500i)6-s − 7-s + (−0.809 − 0.587i)8-s + (−0.702 + 2.16i)9-s + (−1.83 − 1.27i)10-s + (0.452 + 1.39i)11-s + (−0.263 + 0.810i)12-s + (−1.74 + 5.36i)13-s + (−0.309 − 0.951i)14-s + (−0.553 + 1.82i)15-s + (0.309 − 0.951i)16-s + (5.96 + 4.33i)17-s + ⋯ |
| L(s) = 1 | + (0.218 + 0.672i)2-s + (0.397 − 0.289i)3-s + (−0.404 + 0.293i)4-s + (−0.797 + 0.603i)5-s + (0.281 + 0.204i)6-s − 0.377·7-s + (−0.286 − 0.207i)8-s + (−0.234 + 0.721i)9-s + (−0.579 − 0.404i)10-s + (0.136 + 0.419i)11-s + (−0.0759 + 0.233i)12-s + (−0.483 + 1.48i)13-s + (−0.0825 − 0.254i)14-s + (−0.143 + 0.470i)15-s + (0.0772 − 0.237i)16-s + (1.44 + 1.05i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.666 - 0.745i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.666 - 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.466614 + 1.04289i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.466614 + 1.04289i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.309 - 0.951i)T \) |
| 5 | \( 1 + (1.78 - 1.34i)T \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 + (-0.689 + 0.500i)T + (0.927 - 2.85i)T^{2} \) |
| 11 | \( 1 + (-0.452 - 1.39i)T + (-8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (1.74 - 5.36i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (-5.96 - 4.33i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (3.37 + 2.45i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (2.02 + 6.22i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (-0.541 + 0.393i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-6.05 - 4.39i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-0.122 + 0.377i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-3.02 + 9.29i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 3.04T + 43T^{2} \) |
| 47 | \( 1 + (3.95 - 2.87i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (3.80 - 2.76i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-3.92 + 12.0i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-4.12 - 12.7i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (-0.479 - 0.348i)T + (20.7 + 63.7i)T^{2} \) |
| 71 | \( 1 + (-7.08 + 5.14i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-3.04 - 9.37i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-4.33 + 3.15i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-9.51 - 6.91i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (-2.44 - 7.51i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (2.44 - 1.77i)T + (29.9 - 92.2i)T^{2} \) |
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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
−12.10439815190045984998194251417, −10.88495643674947158363371231193, −9.959878111161396258377021906924, −8.671205731925626538706384771783, −7.991562056133801854106407942431, −7.02701762311544163529016615155, −6.37041171156089547744124687531, −4.79868655079081774860280086442, −3.80455833799878570379930857284, −2.40029584567277034231188275543,
0.71846620251379559860180196175, 3.05027569814749039547258699864, 3.65643001950726475214778092534, 4.98925579856032146561372249409, 6.02729790397007597401468863999, 7.71912158450314031151238263915, 8.366179770524156646539968103049, 9.585222295476759747879543386951, 10.00955606685025112440791500133, 11.39134522332891149591287830047
