| L(s) = 1 | + (−1.23 − 3.80i)2-s + (−12.9 + 9.40i)4-s + (−68.3 − 22.1i)5-s + (−35.8 − 49.3i)7-s + (51.7 + 37.6i)8-s + 287. i·10-s + (320. + 241. i)11-s + (−765. + 248. i)13-s + (−143. + 197. i)14-s + (79.1 − 243. i)16-s + (−274. + 844. i)17-s + (−1.22e3 + 1.68e3i)19-s + (1.09e3 − 355. i)20-s + (522. − 1.51e3i)22-s − 3.82e3i·23-s + ⋯ |
| L(s) = 1 | + (−0.218 − 0.672i)2-s + (−0.404 + 0.293i)4-s + (−1.22 − 0.397i)5-s + (−0.276 − 0.380i)7-s + (0.286 + 0.207i)8-s + 0.908i·10-s + (0.798 + 0.601i)11-s + (−1.25 + 0.408i)13-s + (−0.195 + 0.269i)14-s + (0.0772 − 0.237i)16-s + (−0.230 + 0.709i)17-s + (−0.779 + 1.07i)19-s + (0.611 − 0.198i)20-s + (0.230 − 0.668i)22-s − 1.50i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.819 + 0.573i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.819 + 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.9100285531\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9100285531\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.23 + 3.80i)T \) |
| 3 | \( 1 \) |
| 11 | \( 1 + (-320. - 241. i)T \) |
| good | 5 | \( 1 + (68.3 + 22.1i)T + (2.52e3 + 1.83e3i)T^{2} \) |
| 7 | \( 1 + (35.8 + 49.3i)T + (-5.19e3 + 1.59e4i)T^{2} \) |
| 13 | \( 1 + (765. - 248. i)T + (3.00e5 - 2.18e5i)T^{2} \) |
| 17 | \( 1 + (274. - 844. i)T + (-1.14e6 - 8.34e5i)T^{2} \) |
| 19 | \( 1 + (1.22e3 - 1.68e3i)T + (-7.65e5 - 2.35e6i)T^{2} \) |
| 23 | \( 1 + 3.82e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + (-5.04e3 + 3.66e3i)T + (6.33e6 - 1.95e7i)T^{2} \) |
| 31 | \( 1 + (1.10e3 + 3.40e3i)T + (-2.31e7 + 1.68e7i)T^{2} \) |
| 37 | \( 1 + (-3.63e3 + 2.64e3i)T + (2.14e7 - 6.59e7i)T^{2} \) |
| 41 | \( 1 + (-1.60e4 - 1.16e4i)T + (3.58e7 + 1.10e8i)T^{2} \) |
| 43 | \( 1 - 1.10e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + (-2.51e3 + 3.46e3i)T + (-7.08e7 - 2.18e8i)T^{2} \) |
| 53 | \( 1 + (-2.90e4 + 9.44e3i)T + (3.38e8 - 2.45e8i)T^{2} \) |
| 59 | \( 1 + (-1.46e4 - 2.01e4i)T + (-2.20e8 + 6.79e8i)T^{2} \) |
| 61 | \( 1 + (1.74e4 + 5.67e3i)T + (6.83e8 + 4.96e8i)T^{2} \) |
| 67 | \( 1 - 6.44e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (5.26e4 + 1.71e4i)T + (1.45e9 + 1.06e9i)T^{2} \) |
| 73 | \( 1 + (1.13e4 + 1.56e4i)T + (-6.40e8 + 1.97e9i)T^{2} \) |
| 79 | \( 1 + (-1.49e4 + 4.86e3i)T + (2.48e9 - 1.80e9i)T^{2} \) |
| 83 | \( 1 + (-1.74e4 + 5.38e4i)T + (-3.18e9 - 2.31e9i)T^{2} \) |
| 89 | \( 1 - 1.14e5iT - 5.58e9T^{2} \) |
| 97 | \( 1 + (1.72e4 + 5.32e4i)T + (-6.94e9 + 5.04e9i)T^{2} \) |
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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
−11.71479443554253666035784177594, −10.51197172050784337889145197158, −9.658126320622323692791086631531, −8.493589582322867896193270485348, −7.66876164947117787115884398228, −6.47103084138370513180960160083, −4.41860060103483166309546547861, −4.06444884407901132321828344918, −2.31153608911118818728205015580, −0.65365989841581660804039499429,
0.54233962196729740184455231162, 2.86163802544662081530144303639, 4.17173521989061282165356579638, 5.42608879342668766465777529929, 6.84973368036636599854003078209, 7.43743446630308068444250204515, 8.628340488996721469902534747473, 9.440458424337631614783608180296, 10.77662912857649482671839198707, 11.69322832143048231581255216925
