| L(s) = 1 | + (−1.20 + 0.0574i)2-s + (−3.19 + 4.10i)3-s + (−6.51 + 0.621i)4-s + (−10.2 − 9.80i)5-s + (3.61 − 5.12i)6-s + (−0.946 − 4.91i)7-s + (17.3 − 2.49i)8-s + (−6.62 − 26.1i)9-s + (12.9 + 11.2i)10-s + (−7.00 + 3.60i)11-s + (18.2 − 28.6i)12-s + (−78.3 − 15.1i)13-s + (1.42 + 5.86i)14-s + (73.0 − 10.8i)15-s + (30.5 − 5.89i)16-s + (−1.14 − 2.50i)17-s + ⋯ |
| L(s) = 1 | + (−0.426 + 0.0203i)2-s + (−0.614 + 0.789i)3-s + (−0.814 + 0.0777i)4-s + (−0.919 − 0.876i)5-s + (0.245 − 0.348i)6-s + (−0.0511 − 0.265i)7-s + (0.768 − 0.110i)8-s + (−0.245 − 0.969i)9-s + (0.409 + 0.355i)10-s + (−0.191 + 0.0989i)11-s + (0.438 − 0.690i)12-s + (−1.67 − 0.322i)13-s + (0.0271 + 0.112i)14-s + (1.25 − 0.187i)15-s + (0.477 − 0.0920i)16-s + (−0.0163 − 0.0357i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.645 - 0.763i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.645 - 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.367630 + 0.170524i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.367630 + 0.170524i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (3.19 - 4.10i)T \) |
| 23 | \( 1 + (-17.4 - 108. i)T \) |
| good | 2 | \( 1 + (1.20 - 0.0574i)T + (7.96 - 0.760i)T^{2} \) |
| 5 | \( 1 + (10.2 + 9.80i)T + (5.94 + 124. i)T^{2} \) |
| 7 | \( 1 + (0.946 + 4.91i)T + (-318. + 127. i)T^{2} \) |
| 11 | \( 1 + (7.00 - 3.60i)T + (772. - 1.08e3i)T^{2} \) |
| 13 | \( 1 + (78.3 + 15.1i)T + (2.03e3 + 816. i)T^{2} \) |
| 17 | \( 1 + (1.14 + 2.50i)T + (-3.21e3 + 3.71e3i)T^{2} \) |
| 19 | \( 1 + (-29.7 - 13.5i)T + (4.49e3 + 5.18e3i)T^{2} \) |
| 29 | \( 1 + (10.1 - 106. i)T + (-2.39e4 - 4.61e3i)T^{2} \) |
| 31 | \( 1 + (12.6 - 9.93i)T + (7.02e3 - 2.89e4i)T^{2} \) |
| 37 | \( 1 + (4.93 - 16.7i)T + (-4.26e4 - 2.73e4i)T^{2} \) |
| 41 | \( 1 + (113. - 118. i)T + (-3.27e3 - 6.88e4i)T^{2} \) |
| 43 | \( 1 + (4.28 - 5.45i)T + (-1.87e4 - 7.72e4i)T^{2} \) |
| 47 | \( 1 + (-547. - 315. i)T + (5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (228. + 263. i)T + (-2.11e4 + 1.47e5i)T^{2} \) |
| 59 | \( 1 + (-66.5 + 345. i)T + (-1.90e5 - 7.63e4i)T^{2} \) |
| 61 | \( 1 + (-167. + 417. i)T + (-1.64e5 - 1.56e5i)T^{2} \) |
| 67 | \( 1 + (-319. + 619. i)T + (-1.74e5 - 2.44e5i)T^{2} \) |
| 71 | \( 1 + (-322. - 501. i)T + (-1.48e5 + 3.25e5i)T^{2} \) |
| 73 | \( 1 + (-243. + 534. i)T + (-2.54e5 - 2.93e5i)T^{2} \) |
| 79 | \( 1 + (-182. + 62.9i)T + (3.87e5 - 3.04e5i)T^{2} \) |
| 83 | \( 1 + (46.5 - 44.3i)T + (2.72e4 - 5.71e5i)T^{2} \) |
| 89 | \( 1 + (194. - 1.35e3i)T + (-6.76e5 - 1.98e5i)T^{2} \) |
| 97 | \( 1 + (1.28e3 + 311. i)T + (8.11e5 + 4.18e5i)T^{2} \) |
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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
−12.17858382365084406358342068491, −10.99979526497733956591855145031, −9.894782815900995797919134173213, −9.318694502414315658495417061030, −8.190187730397247423271765165749, −7.27778613925872239078876863794, −5.28208427169230908425762096788, −4.70565666451841244712762637791, −3.62532329302260641880083488611, −0.66189179372377956436893388675,
0.40589032392750194922780844361, 2.50916819339246318905749613806, 4.33050085306265213119189673848, 5.50518458612804573280405791682, 7.03821017758654670741870987261, 7.58311179918853789576186001624, 8.674269624995872688440220084059, 9.990582353817117370696053373982, 10.84276263460010169664937963076, 11.87557624927579766466217538065
