| L(s) = 1 | + (−0.322 + 1.37i)2-s + (1.63 − 0.576i)3-s + (−1.79 − 0.889i)4-s + (1.79 + 1.33i)5-s + (0.266 + 2.43i)6-s + 3.04i·7-s + (1.80 − 2.17i)8-s + (2.33 − 1.88i)9-s + (−2.41 + 2.03i)10-s + (−0.519 + 1.59i)11-s + (−3.43 − 0.419i)12-s + (−1.88 − 5.80i)13-s + (−4.18 − 0.982i)14-s + (3.69 + 1.14i)15-s + (2.41 + 3.18i)16-s + (0.841 + 1.15i)17-s + ⋯ |
| L(s) = 1 | + (−0.228 + 0.973i)2-s + (0.942 − 0.332i)3-s + (−0.895 − 0.444i)4-s + (0.802 + 0.596i)5-s + (0.108 + 0.994i)6-s + 1.14i·7-s + (0.637 − 0.770i)8-s + (0.778 − 0.627i)9-s + (−0.764 + 0.644i)10-s + (−0.156 + 0.482i)11-s + (−0.992 − 0.121i)12-s + (−0.523 − 1.61i)13-s + (−1.11 − 0.262i)14-s + (0.955 + 0.295i)15-s + (0.604 + 0.796i)16-s + (0.204 + 0.280i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.205 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.205 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.24291 + 1.00890i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.24291 + 1.00890i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.322 - 1.37i)T \) |
| 3 | \( 1 + (-1.63 + 0.576i)T \) |
| 5 | \( 1 + (-1.79 - 1.33i)T \) |
| good | 7 | \( 1 - 3.04iT - 7T^{2} \) |
| 11 | \( 1 + (0.519 - 1.59i)T + (-8.89 - 6.46i)T^{2} \) |
| 13 | \( 1 + (1.88 + 5.80i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (-0.841 - 1.15i)T + (-5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-2.14 - 2.95i)T + (-5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (1.33 - 4.09i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (1.32 - 1.82i)T + (-8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (3.80 + 5.23i)T + (-9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (3.25 + 10.0i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (-8.40 + 2.73i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 2.84iT - 43T^{2} \) |
| 47 | \( 1 + (5.06 + 3.68i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (5.69 - 7.84i)T + (-16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (3.64 + 11.2i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-0.190 + 0.587i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + (1.22 + 1.68i)T + (-20.7 + 63.7i)T^{2} \) |
| 71 | \( 1 + (0.331 + 0.240i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-0.657 + 2.02i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-8.14 + 11.2i)T + (-24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (5.79 - 4.21i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (-0.333 - 0.108i)T + (72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (0.649 + 0.471i)T + (29.9 + 92.2i)T^{2} \) |
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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
−12.44553994053021612911891771570, −10.58230975796756051216145940204, −9.662724529742743067292780708774, −9.136412349923904493963837158527, −7.898045516928728563770568026508, −7.37863836393864953657807580428, −5.99054871158270037691823085768, −5.37531375913645572999408410031, −3.43505728389450286991955260590, −2.05078779193639759307941200854,
1.45482000153911723310418193684, 2.80369531752899381938096467290, 4.18753100029576222861962116952, 4.90085596210714006349099129046, 6.89805886411007069481080595382, 8.066358098948872098897674116220, 9.046451312417437886922538991050, 9.639939876031965401090414781202, 10.39280040061356602226959121451, 11.37921158191867813158671480609
