| L(s) = 1 | + (0.427 − 1.67i)3-s + (−0.0458 − 0.170i)5-s + (−1.17 − 2.03i)7-s + (−2.63 − 1.43i)9-s + (0.0913 − 0.340i)11-s + (0.399 + 1.49i)13-s + (−0.306 + 0.00382i)15-s − 3.58i·17-s + (−5.36 − 5.36i)19-s + (−3.91 + 1.10i)21-s + (−0.165 − 0.0953i)23-s + (4.30 − 2.48i)25-s + (−3.53 + 3.80i)27-s + (−2.43 + 9.10i)29-s + (−3.43 − 1.98i)31-s + ⋯ |
| L(s) = 1 | + (0.246 − 0.969i)3-s + (−0.0204 − 0.0764i)5-s + (−0.443 − 0.768i)7-s + (−0.878 − 0.478i)9-s + (0.0275 − 0.102i)11-s + (0.110 + 0.413i)13-s + (−0.0791 + 0.000988i)15-s − 0.868i·17-s + (−1.23 − 1.23i)19-s + (−0.854 + 0.240i)21-s + (−0.0344 − 0.0198i)23-s + (0.860 − 0.496i)25-s + (−0.680 + 0.733i)27-s + (−0.452 + 1.69i)29-s + (−0.617 − 0.356i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.769 + 0.638i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.769 + 0.638i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.388408 - 1.07593i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.388408 - 1.07593i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.427 + 1.67i)T \) |
| good | 5 | \( 1 + (0.0458 + 0.170i)T + (-4.33 + 2.5i)T^{2} \) |
| 7 | \( 1 + (1.17 + 2.03i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.0913 + 0.340i)T + (-9.52 - 5.5i)T^{2} \) |
| 13 | \( 1 + (-0.399 - 1.49i)T + (-11.2 + 6.5i)T^{2} \) |
| 17 | \( 1 + 3.58iT - 17T^{2} \) |
| 19 | \( 1 + (5.36 + 5.36i)T + 19iT^{2} \) |
| 23 | \( 1 + (0.165 + 0.0953i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (2.43 - 9.10i)T + (-25.1 - 14.5i)T^{2} \) |
| 31 | \( 1 + (3.43 + 1.98i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3.28 + 3.28i)T + 37iT^{2} \) |
| 41 | \( 1 + (-4.25 + 7.37i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.09 - 1.09i)T + (37.2 + 21.5i)T^{2} \) |
| 47 | \( 1 + (-4.93 - 8.53i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.83 + 4.83i)T - 53iT^{2} \) |
| 59 | \( 1 + (-2.68 + 0.720i)T + (51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (7.97 + 2.13i)T + (52.8 + 30.5i)T^{2} \) |
| 67 | \( 1 + (11.4 - 3.06i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 - 1.13iT - 71T^{2} \) |
| 73 | \( 1 + 5.67iT - 73T^{2} \) |
| 79 | \( 1 + (-12.8 + 7.42i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-12.3 - 3.31i)T + (71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + 3.05T + 89T^{2} \) |
| 97 | \( 1 + (0.996 + 1.72i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.61197271101100922520693985046, −9.138598027866464519197536854826, −8.802891343242643251780061542309, −7.37844408053870823169626024554, −7.01736908919818669035952308573, −6.04583878548493128068806189599, −4.71269455356004737715356676442, −3.44166959473636057192440469997, −2.23556737363903290752247014418, −0.60838022626162135068420152669,
2.24108988634808591697441503999, 3.44098373339593491268142897822, 4.35227388523480307285147481407, 5.63240080703539929660874018376, 6.22415412467884086355588987076, 7.76816568851204026148756944978, 8.603692059519650053682575623553, 9.288776500991284865357782371781, 10.27772330157513393828592521034, 10.75546790147094823202502990573
