| L(s) = 1 | + (0.425 − 1.58i)2-s + (−9.75 + 2.61i)3-s + (25.3 + 14.6i)4-s + (−51.8 − 20.8i)5-s + 16.5i·6-s + (−111. + 65.3i)7-s + (71.1 − 71.1i)8-s + (−122. + 70.4i)9-s + (−55.1 + 73.4i)10-s + (−330. + 571. i)11-s + (−285. − 76.6i)12-s + (67.3 + 67.3i)13-s + (56.0 + 205. i)14-s + (560. + 67.9i)15-s + (386. + 668. i)16-s + (−388. − 1.45e3i)17-s + ⋯ |
| L(s) = 1 | + (0.0751 − 0.280i)2-s + (−0.625 + 0.167i)3-s + (0.793 + 0.457i)4-s + (−0.927 − 0.373i)5-s + 0.188i·6-s + (−0.863 + 0.504i)7-s + (0.393 − 0.393i)8-s + (−0.502 + 0.290i)9-s + (−0.174 + 0.232i)10-s + (−0.822 + 1.42i)11-s + (−0.573 − 0.153i)12-s + (0.110 + 0.110i)13-s + (0.0764 + 0.280i)14-s + (0.643 + 0.0779i)15-s + (0.377 + 0.653i)16-s + (−0.326 − 1.21i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.735 - 0.677i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.735 - 0.677i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.185104 + 0.473997i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.185104 + 0.473997i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (51.8 + 20.8i)T \) |
| 7 | \( 1 + (111. - 65.3i)T \) |
| good | 2 | \( 1 + (-0.425 + 1.58i)T + (-27.7 - 16i)T^{2} \) |
| 3 | \( 1 + (9.75 - 2.61i)T + (210. - 121.5i)T^{2} \) |
| 11 | \( 1 + (330. - 571. i)T + (-8.05e4 - 1.39e5i)T^{2} \) |
| 13 | \( 1 + (-67.3 - 67.3i)T + 3.71e5iT^{2} \) |
| 17 | \( 1 + (388. + 1.45e3i)T + (-1.22e6 + 7.09e5i)T^{2} \) |
| 19 | \( 1 + (113. + 196. i)T + (-1.23e6 + 2.14e6i)T^{2} \) |
| 23 | \( 1 + (772. + 206. i)T + (5.57e6 + 3.21e6i)T^{2} \) |
| 29 | \( 1 + 3.42e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + (-3.63e3 - 2.10e3i)T + (1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 + (1.38e3 - 5.15e3i)T + (-6.00e7 - 3.46e7i)T^{2} \) |
| 41 | \( 1 - 1.42e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 + (1.16e4 - 1.16e4i)T - 1.47e8iT^{2} \) |
| 47 | \( 1 + (1.35e4 + 3.63e3i)T + (1.98e8 + 1.14e8i)T^{2} \) |
| 53 | \( 1 + (3.32e3 + 1.24e4i)T + (-3.62e8 + 2.09e8i)T^{2} \) |
| 59 | \( 1 + (1.26e4 - 2.18e4i)T + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (2.96e4 - 1.71e4i)T + (4.22e8 - 7.31e8i)T^{2} \) |
| 67 | \( 1 + (-4.72e4 + 1.26e4i)T + (1.16e9 - 6.75e8i)T^{2} \) |
| 71 | \( 1 - 7.10e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + (2.88e4 - 7.73e3i)T + (1.79e9 - 1.03e9i)T^{2} \) |
| 79 | \( 1 + (6.22e4 - 3.59e4i)T + (1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 + (-3.31e4 - 3.31e4i)T + 3.93e9iT^{2} \) |
| 89 | \( 1 + (5.66e4 + 9.80e4i)T + (-2.79e9 + 4.83e9i)T^{2} \) |
| 97 | \( 1 + (1.04e5 - 1.04e5i)T - 8.58e9iT^{2} \) |
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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
−15.96682284212972916558719591552, −15.31359662546664379726051449496, −13.09703837178915816816726158239, −12.09137491521619919360201723000, −11.39116924537284956735761429578, −9.935122768890413888857042459469, −8.054066700836824207446272298445, −6.70126914092614628919687073159, −4.78634981149971649260687728391, −2.77143148629240435853805958739,
0.29295041046471234333784347055, 3.30625094477908674542381603720, 5.78706501620947685782265272319, 6.75891912263564132606382258797, 8.240629252999275203261911748507, 10.55991498384488083680005062969, 11.14317884314852239889975435444, 12.45147485182981434983404498482, 14.00821885978351250637272311137, 15.36251651388759390140281394745
