| L(s) = 1 | + (0.146 + 3.07i)2-s + (−2.53 + 4.53i)3-s + (−1.49 + 0.143i)4-s + (−7.20 − 8.31i)5-s + (−14.3 − 7.13i)6-s + (−15.1 + 29.3i)7-s + (2.84 + 19.8i)8-s + (−14.1 − 22.9i)9-s + (24.5 − 23.3i)10-s + (2.92 − 8.45i)11-s + (3.14 − 7.16i)12-s + (3.02 − 3.84i)13-s + (−92.6 − 42.3i)14-s + (55.9 − 11.6i)15-s + (−72.4 + 13.9i)16-s + (8.65 − 90.6i)17-s + ⋯ |
| L(s) = 1 | + (0.0518 + 1.08i)2-s + (−0.487 + 0.873i)3-s + (−0.187 + 0.0178i)4-s + (−0.644 − 0.743i)5-s + (−0.976 − 0.485i)6-s + (−0.817 + 1.58i)7-s + (0.125 + 0.875i)8-s + (−0.525 − 0.851i)9-s + (0.775 − 0.739i)10-s + (0.0801 − 0.231i)11-s + (0.0756 − 0.172i)12-s + (0.0645 − 0.0820i)13-s + (−1.76 − 0.807i)14-s + (0.962 − 0.200i)15-s + (−1.13 + 0.218i)16-s + (0.123 − 1.29i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.287 + 0.957i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.287 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.192470 - 0.143180i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.192470 - 0.143180i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.53 - 4.53i)T \) |
| 67 | \( 1 + (396. + 378. i)T \) |
| good | 2 | \( 1 + (-0.146 - 3.07i)T + (-7.96 + 0.760i)T^{2} \) |
| 5 | \( 1 + (7.20 + 8.31i)T + (-17.7 + 123. i)T^{2} \) |
| 7 | \( 1 + (15.1 - 29.3i)T + (-198. - 279. i)T^{2} \) |
| 11 | \( 1 + (-2.92 + 8.45i)T + (-1.04e3 - 822. i)T^{2} \) |
| 13 | \( 1 + (-3.02 + 3.84i)T + (-517. - 2.13e3i)T^{2} \) |
| 17 | \( 1 + (-8.65 + 90.6i)T + (-4.82e3 - 929. i)T^{2} \) |
| 19 | \( 1 + (-7.16 + 3.69i)T + (3.97e3 - 5.58e3i)T^{2} \) |
| 23 | \( 1 + (23.9 + 5.81i)T + (1.08e4 + 5.57e3i)T^{2} \) |
| 29 | \( 1 + (119. - 69.0i)T + (1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (38.6 + 49.1i)T + (-7.02e3 + 2.89e4i)T^{2} \) |
| 37 | \( 1 + (4.00 - 6.93i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (12.9 - 18.2i)T + (-2.25e4 - 6.51e4i)T^{2} \) |
| 43 | \( 1 + (-444. + 203. i)T + (5.20e4 - 6.00e4i)T^{2} \) |
| 47 | \( 1 + (330. - 347. i)T + (-4.94e3 - 1.03e5i)T^{2} \) |
| 53 | \( 1 + (234. - 514. i)T + (-9.74e4 - 1.12e5i)T^{2} \) |
| 59 | \( 1 + (667. - 95.9i)T + (1.97e5 - 5.78e4i)T^{2} \) |
| 61 | \( 1 + (586. - 202. i)T + (1.78e5 - 1.40e5i)T^{2} \) |
| 71 | \( 1 + (32.4 + 339. i)T + (-3.51e5 + 6.77e4i)T^{2} \) |
| 73 | \( 1 + (130. + 378. i)T + (-3.05e5 + 2.40e5i)T^{2} \) |
| 79 | \( 1 + (-454. - 1.13e3i)T + (-3.56e5 + 3.40e5i)T^{2} \) |
| 83 | \( 1 + (50.9 + 264. i)T + (-5.30e5 + 2.12e5i)T^{2} \) |
| 89 | \( 1 + (-267. + 912. i)T + (-5.93e5 - 3.81e5i)T^{2} \) |
| 97 | \( 1 + (688. + 397. i)T + (4.56e5 + 7.90e5i)T^{2} \) |
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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
−12.47398894644454317907199451001, −11.90605464633917406790669572476, −10.96769962137444167415791420539, −9.318364031501329176385643379087, −8.948437187181210087418848786856, −7.71338587492199935005231242850, −6.28155811827206760959982106956, −5.57933512201560096560327546945, −4.64886387973888147385304279348, −2.93810627956841436662711828550,
0.10623196663079094611811381663, 1.54823220401083508534493043037, 3.21230732554820787177575205552, 4.10829183893235646504754204671, 6.29557162989799344121889749938, 7.07093464074168278445787248663, 7.80225364592933994528387694364, 9.746385315881610799748227245281, 10.73853287918531997737244548105, 11.04218807737002868285812497657
