| L(s) = 1 | − 1.05·2-s + (31.1 − 17.9i)3-s − 62.8·4-s + (−116. + 201. i)5-s + (−32.8 + 18.9i)6-s + (166. + 288. i)7-s + 133.·8-s + (281. − 487. i)9-s + (122. − 212. i)10-s + (1.17e3 + 680. i)11-s + (−1.95e3 + 1.13e3i)12-s + (−422. − 244. i)13-s + (−175. − 304. i)14-s + 8.35e3i·15-s + 3.88e3·16-s + (−8.15e3 + 4.70e3i)17-s + ⋯ |
| L(s) = 1 | − 0.131·2-s + (1.15 − 0.665i)3-s − 0.982·4-s + (−0.929 + 1.61i)5-s + (−0.151 + 0.0877i)6-s + (0.485 + 0.841i)7-s + 0.261·8-s + (0.385 − 0.668i)9-s + (0.122 − 0.212i)10-s + (0.885 + 0.511i)11-s + (−1.13 + 0.653i)12-s + (−0.192 − 0.111i)13-s + (−0.0640 − 0.110i)14-s + 2.47i·15-s + 0.948·16-s + (−1.65 + 0.957i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0207 - 0.999i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.0207 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.953780 + 0.934220i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.953780 + 0.934220i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 31 | \( 1 + (-2.09e4 - 2.11e4i)T \) |
| good | 2 | \( 1 + 1.05T + 64T^{2} \) |
| 3 | \( 1 + (-31.1 + 17.9i)T + (364.5 - 631. i)T^{2} \) |
| 5 | \( 1 + (116. - 201. i)T + (-7.81e3 - 1.35e4i)T^{2} \) |
| 7 | \( 1 + (-166. - 288. i)T + (-5.88e4 + 1.01e5i)T^{2} \) |
| 11 | \( 1 + (-1.17e3 - 680. i)T + (8.85e5 + 1.53e6i)T^{2} \) |
| 13 | \( 1 + (422. + 244. i)T + (2.41e6 + 4.18e6i)T^{2} \) |
| 17 | \( 1 + (8.15e3 - 4.70e3i)T + (1.20e7 - 2.09e7i)T^{2} \) |
| 19 | \( 1 + (1.49e3 + 2.58e3i)T + (-2.35e7 + 4.07e7i)T^{2} \) |
| 23 | \( 1 + 1.73e3iT - 1.48e8T^{2} \) |
| 29 | \( 1 - 9.33e3iT - 5.94e8T^{2} \) |
| 37 | \( 1 + (-5.22e4 + 3.01e4i)T + (1.28e9 - 2.22e9i)T^{2} \) |
| 41 | \( 1 + (-3.72e4 + 6.44e4i)T + (-2.37e9 - 4.11e9i)T^{2} \) |
| 43 | \( 1 + (5.35e4 - 3.09e4i)T + (3.16e9 - 5.47e9i)T^{2} \) |
| 47 | \( 1 - 1.32e5T + 1.07e10T^{2} \) |
| 53 | \( 1 + (-1.83e4 - 1.05e4i)T + (1.10e10 + 1.91e10i)T^{2} \) |
| 59 | \( 1 + (-1.26e5 - 2.19e5i)T + (-2.10e10 + 3.65e10i)T^{2} \) |
| 61 | \( 1 + 8.80e4iT - 5.15e10T^{2} \) |
| 67 | \( 1 + (-4.14e4 + 7.17e4i)T + (-4.52e10 - 7.83e10i)T^{2} \) |
| 71 | \( 1 + (-1.74e5 + 3.02e5i)T + (-6.40e10 - 1.10e11i)T^{2} \) |
| 73 | \( 1 + (-1.26e5 - 7.31e4i)T + (7.56e10 + 1.31e11i)T^{2} \) |
| 79 | \( 1 + (6.73e4 - 3.89e4i)T + (1.21e11 - 2.10e11i)T^{2} \) |
| 83 | \( 1 + (-7.95e5 - 4.59e5i)T + (1.63e11 + 2.83e11i)T^{2} \) |
| 89 | \( 1 - 9.63e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 1.64e6T + 8.32e11T^{2} \) |
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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
−15.16853325274437522930432307481, −14.76357862680980468801856555891, −13.76276864107667869709527848871, −12.33123839683352237575039787164, −10.84602434816728184541315251337, −9.028019097476363653705041907568, −8.087473815242045513665217823689, −6.81163664231195984621165526321, −4.01897860786743879344407304392, −2.39027516896525371221995712855,
0.70717600269382981759797785118, 4.04007388774093183466131249524, 4.55574900963431189514371014093, 7.962453512132956295375095146736, 8.774646071280309704919270565831, 9.546808916354064468711536179610, 11.58025221879775309312063185300, 13.20899813837926349195559115753, 14.00564106587694912004757132759, 15.28221571556578829124572505090
