| L(s) = 1 | + (−9.23 − 5.33i)2-s + (−22.3 − 15.0i)3-s + (24.9 + 43.1i)4-s + (−152. − 88.2i)5-s + (126. + 258. i)6-s + (342. − 17.4i)7-s + 151. i·8-s + (274. + 675. i)9-s + (941. + 1.63e3i)10-s + (766. − 442. i)11-s + (92.8 − 1.34e3i)12-s − 3.03e3·13-s + (−3.25e3 − 1.66e3i)14-s + (2.09e3 + 4.28e3i)15-s + (2.40e3 − 4.15e3i)16-s + (408. − 235. i)17-s + ⋯ |
| L(s) = 1 | + (−1.15 − 0.666i)2-s + (−0.829 − 0.558i)3-s + (0.389 + 0.674i)4-s + (−1.22 − 0.705i)5-s + (0.585 + 1.19i)6-s + (0.998 − 0.0508i)7-s + 0.295i·8-s + (0.375 + 0.926i)9-s + (0.941 + 1.63i)10-s + (0.575 − 0.332i)11-s + (0.0537 − 0.776i)12-s − 1.38·13-s + (−1.18 − 0.607i)14-s + (0.619 + 1.26i)15-s + (0.586 − 1.01i)16-s + (0.0831 − 0.0480i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.460 - 0.887i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.460 - 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.0910363 + 0.0553336i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0910363 + 0.0553336i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (22.3 + 15.0i)T \) |
| 7 | \( 1 + (-342. + 17.4i)T \) |
| good | 2 | \( 1 + (9.23 + 5.33i)T + (32 + 55.4i)T^{2} \) |
| 5 | \( 1 + (152. + 88.2i)T + (7.81e3 + 1.35e4i)T^{2} \) |
| 11 | \( 1 + (-766. + 442. i)T + (8.85e5 - 1.53e6i)T^{2} \) |
| 13 | \( 1 + 3.03e3T + 4.82e6T^{2} \) |
| 17 | \( 1 + (-408. + 235. i)T + (1.20e7 - 2.09e7i)T^{2} \) |
| 19 | \( 1 + (4.20e3 - 7.28e3i)T + (-2.35e7 - 4.07e7i)T^{2} \) |
| 23 | \( 1 + (489. + 282. i)T + (7.40e7 + 1.28e8i)T^{2} \) |
| 29 | \( 1 - 3.50e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 + (-1.29e4 - 2.24e4i)T + (-4.43e8 + 7.68e8i)T^{2} \) |
| 37 | \( 1 + (-1.75e3 + 3.03e3i)T + (-1.28e9 - 2.22e9i)T^{2} \) |
| 41 | \( 1 - 1.16e3iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 9.18e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + (4.77e4 + 2.75e4i)T + (5.38e9 + 9.33e9i)T^{2} \) |
| 53 | \( 1 + (-1.48e4 + 8.54e3i)T + (1.10e10 - 1.91e10i)T^{2} \) |
| 59 | \( 1 + (3.27e5 - 1.89e5i)T + (2.10e10 - 3.65e10i)T^{2} \) |
| 61 | \( 1 + (3.94e4 - 6.83e4i)T + (-2.57e10 - 4.46e10i)T^{2} \) |
| 67 | \( 1 + (-3.01e4 - 5.22e4i)T + (-4.52e10 + 7.83e10i)T^{2} \) |
| 71 | \( 1 - 3.30e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + (1.82e5 + 3.15e5i)T + (-7.56e10 + 1.31e11i)T^{2} \) |
| 79 | \( 1 + (-1.83e5 + 3.18e5i)T + (-1.21e11 - 2.10e11i)T^{2} \) |
| 83 | \( 1 + 5.18e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + (6.51e5 + 3.76e5i)T + (2.48e11 + 4.30e11i)T^{2} \) |
| 97 | \( 1 - 1.02e6T + 8.32e11T^{2} \) |
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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
−17.20117678596589807686146288047, −16.49372648269927978148499335975, −14.55620077552879473190571379790, −12.25923672626955968422805309290, −11.72041342393887810561051577663, −10.49409247776680844058457606254, −8.557023291600456219241505027225, −7.54889360392251817824400462498, −4.88840728004898527098663628174, −1.39993726423776346994916506625,
0.11838417123470289921276382739, 4.40155159890011055037474696300, 6.80452531315571609317886629115, 7.937411009553097603602724525613, 9.643870776348017847885162427472, 11.04288218706457312809470159022, 12.04838391785358843544210114371, 14.96502383330365941657031435789, 15.39619806655299310096325806839, 16.93377306263225818250129817909
