| L(s) = 1 | + (1.61 + 1.17i)2-s + (−2.41 + 7.42i)3-s + (1.23 + 3.80i)4-s + (−12.0 + 8.76i)5-s + (−12.6 + 9.18i)6-s + (6.72 + 20.7i)7-s + (−2.47 + 7.60i)8-s + (−27.5 − 19.9i)9-s − 29.8·10-s − 31.2·12-s + (35.6 + 25.8i)13-s + (−13.4 + 41.4i)14-s + (−36.0 − 110. i)15-s + (−12.9 + 9.40i)16-s + (20.1 − 14.6i)17-s + (−21.0 − 64.7i)18-s + ⋯ |
| L(s) = 1 | + (0.572 + 0.415i)2-s + (−0.464 + 1.42i)3-s + (0.154 + 0.475i)4-s + (−1.07 + 0.784i)5-s + (−0.860 + 0.624i)6-s + (0.363 + 1.11i)7-s + (−0.109 + 0.336i)8-s + (−1.01 − 0.740i)9-s − 0.943·10-s − 0.751·12-s + (0.759 + 0.552i)13-s + (−0.256 + 0.790i)14-s + (−0.619 − 1.90i)15-s + (−0.202 + 0.146i)16-s + (0.287 − 0.208i)17-s + (−0.275 − 0.847i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 242 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 + 0.766i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 242 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.648073 - 1.38929i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.648073 - 1.38929i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.61 - 1.17i)T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + (2.41 - 7.42i)T + (-21.8 - 15.8i)T^{2} \) |
| 5 | \( 1 + (12.0 - 8.76i)T + (38.6 - 118. i)T^{2} \) |
| 7 | \( 1 + (-6.72 - 20.7i)T + (-277. + 201. i)T^{2} \) |
| 13 | \( 1 + (-35.6 - 25.8i)T + (678. + 2.08e3i)T^{2} \) |
| 17 | \( 1 + (-20.1 + 14.6i)T + (1.51e3 - 4.67e3i)T^{2} \) |
| 19 | \( 1 + (-6.78 + 20.8i)T + (-5.54e3 - 4.03e3i)T^{2} \) |
| 23 | \( 1 - 177.T + 1.21e4T^{2} \) |
| 29 | \( 1 + (46.1 + 142. i)T + (-1.97e4 + 1.43e4i)T^{2} \) |
| 31 | \( 1 + (60.7 + 44.1i)T + (9.20e3 + 2.83e4i)T^{2} \) |
| 37 | \( 1 + (-68.7 - 211. i)T + (-4.09e4 + 2.97e4i)T^{2} \) |
| 41 | \( 1 + (78.2 - 240. i)T + (-5.57e4 - 4.05e4i)T^{2} \) |
| 43 | \( 1 - 130.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-154. + 474. i)T + (-8.39e4 - 6.10e4i)T^{2} \) |
| 53 | \( 1 + (10.4 + 7.60i)T + (4.60e4 + 1.41e5i)T^{2} \) |
| 59 | \( 1 + (-10.9 - 33.8i)T + (-1.66e5 + 1.20e5i)T^{2} \) |
| 61 | \( 1 + (435. - 316. i)T + (7.01e4 - 2.15e5i)T^{2} \) |
| 67 | \( 1 + 519.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (63.4 - 46.1i)T + (1.10e5 - 3.40e5i)T^{2} \) |
| 73 | \( 1 + (-353. - 1.08e3i)T + (-3.14e5 + 2.28e5i)T^{2} \) |
| 79 | \( 1 + (625. + 454. i)T + (1.52e5 + 4.68e5i)T^{2} \) |
| 83 | \( 1 + (434. - 315. i)T + (1.76e5 - 5.43e5i)T^{2} \) |
| 89 | \( 1 - 667.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-145. - 105. i)T + (2.82e5 + 8.68e5i)T^{2} \) |
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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
−11.74943202714514096060661440744, −11.51912143389157250939281409650, −10.66038333323074574832251603010, −9.339779827950311641874253521829, −8.436310039416983679206134465198, −7.16862703100608848041313149126, −5.93337003747215409627662965047, −4.93891363600133483393325247540, −3.97617781387884863247953477807, −2.96370688976715080579972796161,
0.60875905526457326674449107027, 1.37686211962384787547635006558, 3.47110082477593793495540802700, 4.65690659872422110053143779107, 5.86740115782727175826573295345, 7.19638382577181177487523451778, 7.70913135501741756560626682494, 8.865899139930832566203720365626, 10.73601585170758002610989089893, 11.19929871654715250130019748295
