| L(s) = 1 | + (1.90 + 0.618i)2-s + (−1.78 + 2.45i)3-s + (3.23 + 2.35i)4-s + (3.46 + 10.6i)5-s + (−4.91 + 3.56i)6-s − 1.93i·7-s + (4.70 + 6.47i)8-s + (5.49 + 16.9i)9-s + (0.0144 + 22.3i)10-s + (12.0 − 37.1i)11-s + (−11.5 + 3.75i)12-s + (6.20 − 2.01i)13-s + (1.19 − 3.68i)14-s + (−32.2 − 10.4i)15-s + (4.94 + 15.2i)16-s + (−69.2 − 95.3i)17-s + ⋯ |
| L(s) = 1 | + (0.672 + 0.218i)2-s + (−0.343 + 0.472i)3-s + (0.404 + 0.293i)4-s + (0.309 + 0.950i)5-s + (−0.334 + 0.242i)6-s − 0.104i·7-s + (0.207 + 0.286i)8-s + (0.203 + 0.626i)9-s + (0.000455 + 0.707i)10-s + (0.330 − 1.01i)11-s + (−0.277 + 0.0902i)12-s + (0.132 − 0.0429i)13-s + (0.0228 − 0.0703i)14-s + (−0.555 − 0.180i)15-s + (0.0772 + 0.237i)16-s + (−0.988 − 1.36i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.366 - 0.930i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.366 - 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.52022 + 1.03457i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.52022 + 1.03457i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.90 - 0.618i)T \) |
| 5 | \( 1 + (-3.46 - 10.6i)T \) |
| good | 3 | \( 1 + (1.78 - 2.45i)T + (-8.34 - 25.6i)T^{2} \) |
| 7 | \( 1 + 1.93iT - 343T^{2} \) |
| 11 | \( 1 + (-12.0 + 37.1i)T + (-1.07e3 - 782. i)T^{2} \) |
| 13 | \( 1 + (-6.20 + 2.01i)T + (1.77e3 - 1.29e3i)T^{2} \) |
| 17 | \( 1 + (69.2 + 95.3i)T + (-1.51e3 + 4.67e3i)T^{2} \) |
| 19 | \( 1 + (-80.0 + 58.1i)T + (2.11e3 - 6.52e3i)T^{2} \) |
| 23 | \( 1 + (-68.5 - 22.2i)T + (9.84e3 + 7.15e3i)T^{2} \) |
| 29 | \( 1 + (29.7 + 21.6i)T + (7.53e3 + 2.31e4i)T^{2} \) |
| 31 | \( 1 + (-144. + 105. i)T + (9.20e3 - 2.83e4i)T^{2} \) |
| 37 | \( 1 + (96.9 - 31.4i)T + (4.09e4 - 2.97e4i)T^{2} \) |
| 41 | \( 1 + (-66.5 - 204. i)T + (-5.57e4 + 4.05e4i)T^{2} \) |
| 43 | \( 1 - 276. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (81.6 - 112. i)T + (-3.20e4 - 9.87e4i)T^{2} \) |
| 53 | \( 1 + (379. - 522. i)T + (-4.60e4 - 1.41e5i)T^{2} \) |
| 59 | \( 1 + (151. + 466. i)T + (-1.66e5 + 1.20e5i)T^{2} \) |
| 61 | \( 1 + (-160. + 493. i)T + (-1.83e5 - 1.33e5i)T^{2} \) |
| 67 | \( 1 + (511. + 704. i)T + (-9.29e4 + 2.86e5i)T^{2} \) |
| 71 | \( 1 + (629. + 457. i)T + (1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 + (273. + 88.9i)T + (3.14e5 + 2.28e5i)T^{2} \) |
| 79 | \( 1 + (-326. - 237. i)T + (1.52e5 + 4.68e5i)T^{2} \) |
| 83 | \( 1 + (349. + 480. i)T + (-1.76e5 + 5.43e5i)T^{2} \) |
| 89 | \( 1 + (405. - 1.24e3i)T + (-5.70e5 - 4.14e5i)T^{2} \) |
| 97 | \( 1 + (-636. + 876. i)T + (-2.82e5 - 8.68e5i)T^{2} \) |
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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
−15.36184468843218321281810906712, −13.92793076134489016444889143220, −13.47111296981918206882056188355, −11.46822481333358995052246574390, −10.95772571653123012047223557745, −9.457497928642354962236625535616, −7.50938903867858031104079414298, −6.22666812075857579258492323748, −4.80685076842558440834168033033, −2.97074752157033150948964921616,
1.54944998517384806615448499725, 4.17163592764203484317622127399, 5.69045383600705177515892168215, 6.98075192716717823667965693285, 8.837228153547693048969676877065, 10.17967904980229278014171523474, 11.85998387905004059285463285552, 12.52759561068434923132069997156, 13.38025810403286639853700301514, 14.77887374535227438694511227948
