L(s) = 1 | + (−13.5 − 7.79i)3-s + (−65.9 + 38.1i)5-s + (334. − 77.8i)7-s + (121.5 + 210. i)9-s + (361. − 626. i)11-s + 681. i·13-s + 1.18e3·15-s + (−645. − 372. i)17-s + (3.27e3 − 1.89e3i)19-s + (−5.11e3 − 1.55e3i)21-s + (1.73e3 + 3.00e3i)23-s + (−4.90e3 + 8.50e3i)25-s − 3.78e3i·27-s + 1.62e4·29-s + (−3.98e4 − 2.30e4i)31-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.288i)3-s + (−0.527 + 0.304i)5-s + (0.973 − 0.226i)7-s + (0.166 + 0.288i)9-s + (0.271 − 0.470i)11-s + 0.310i·13-s + 0.351·15-s + (−0.131 − 0.0758i)17-s + (0.477 − 0.275i)19-s + (−0.552 − 0.167i)21-s + (0.142 + 0.246i)23-s + (−0.314 + 0.544i)25-s − 0.192i·27-s + 0.664·29-s + (−1.33 − 0.772i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.167i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.985 + 0.167i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(1.707083022\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.707083022\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (13.5 + 7.79i)T \) |
| 7 | \( 1 + (-334. + 77.8i)T \) |
good | 5 | \( 1 + (65.9 - 38.1i)T + (7.81e3 - 1.35e4i)T^{2} \) |
| 11 | \( 1 + (-361. + 626. i)T + (-8.85e5 - 1.53e6i)T^{2} \) |
| 13 | \( 1 - 681. iT - 4.82e6T^{2} \) |
| 17 | \( 1 + (645. + 372. i)T + (1.20e7 + 2.09e7i)T^{2} \) |
| 19 | \( 1 + (-3.27e3 + 1.89e3i)T + (2.35e7 - 4.07e7i)T^{2} \) |
| 23 | \( 1 + (-1.73e3 - 3.00e3i)T + (-7.40e7 + 1.28e8i)T^{2} \) |
| 29 | \( 1 - 1.62e4T + 5.94e8T^{2} \) |
| 31 | \( 1 + (3.98e4 + 2.30e4i)T + (4.43e8 + 7.68e8i)T^{2} \) |
| 37 | \( 1 + (-2.99e4 - 5.18e4i)T + (-1.28e9 + 2.22e9i)T^{2} \) |
| 41 | \( 1 + 2.22e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 1.37e5T + 6.32e9T^{2} \) |
| 47 | \( 1 + (-1.74e4 + 1.00e4i)T + (5.38e9 - 9.33e9i)T^{2} \) |
| 53 | \( 1 + (7.46e4 - 1.29e5i)T + (-1.10e10 - 1.91e10i)T^{2} \) |
| 59 | \( 1 + (-2.38e5 - 1.37e5i)T + (2.10e10 + 3.65e10i)T^{2} \) |
| 61 | \( 1 + (2.45e4 - 1.41e4i)T + (2.57e10 - 4.46e10i)T^{2} \) |
| 67 | \( 1 + (-1.83e5 + 3.17e5i)T + (-4.52e10 - 7.83e10i)T^{2} \) |
| 71 | \( 1 - 6.27e5T + 1.28e11T^{2} \) |
| 73 | \( 1 + (-3.62e4 - 2.09e4i)T + (7.56e10 + 1.31e11i)T^{2} \) |
| 79 | \( 1 + (1.70e5 + 2.95e5i)T + (-1.21e11 + 2.10e11i)T^{2} \) |
| 83 | \( 1 - 9.97e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + (-1.11e6 + 6.41e5i)T + (2.48e11 - 4.30e11i)T^{2} \) |
| 97 | \( 1 + 1.34e6iT - 8.32e11T^{2} \) |
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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
−10.85845823976100927551195460727, −9.627753565113711969428808867688, −8.453364996648273988310787502782, −7.59371596564306474110764052183, −6.78337501412157901906980705512, −5.57917566684455103132455072028, −4.58365895387008377870093935232, −3.43230258321002983905118604106, −1.86181309665869193692649625954, −0.69940839274835253977069717131,
0.69364944797848598183906588812, 1.96977008273411030581430529070, 3.64649101105955930103242814574, 4.68878045531864925260570331390, 5.42889361603928488670184554107, 6.71992472631678684858685588595, 7.81431551317012780803020191257, 8.603916349018493663729851657843, 9.695237897695904988488672366060, 10.68722512163131818914704659229