| L(s) = 1 | + (−4.63 − 3.23i)2-s + (11.0 + 30.0i)4-s + (−61.0 + 35.2i)5-s + (181. + 104. i)7-s + (46.1 − 175. i)8-s + (397. + 34.1i)10-s + (−75.3 + 130. i)11-s + (−210. − 365. i)13-s + (−501. − 1.07e3i)14-s + (−780. + 662. i)16-s + 662. i·17-s − 624. i·19-s + (−1.73e3 − 1.44e3i)20-s + (772. − 361. i)22-s + (−1.18e3 − 2.05e3i)23-s + ⋯ |
| L(s) = 1 | + (−0.819 − 0.572i)2-s + (0.344 + 0.938i)4-s + (−1.09 + 0.630i)5-s + (1.39 + 0.806i)7-s + (0.254 − 0.966i)8-s + (1.25 + 0.108i)10-s + (−0.187 + 0.325i)11-s + (−0.346 − 0.599i)13-s + (−0.683 − 1.46i)14-s + (−0.762 + 0.646i)16-s + 0.556i·17-s − 0.396i·19-s + (−0.967 − 0.807i)20-s + (0.340 − 0.159i)22-s + (−0.467 − 0.810i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.907 - 0.420i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.907 - 0.420i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.0646527 + 0.292957i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0646527 + 0.292957i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (4.63 + 3.23i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (61.0 - 35.2i)T + (1.56e3 - 2.70e3i)T^{2} \) |
| 7 | \( 1 + (-181. - 104. i)T + (8.40e3 + 1.45e4i)T^{2} \) |
| 11 | \( 1 + (75.3 - 130. i)T + (-8.05e4 - 1.39e5i)T^{2} \) |
| 13 | \( 1 + (210. + 365. i)T + (-1.85e5 + 3.21e5i)T^{2} \) |
| 17 | \( 1 - 662. iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 624. iT - 2.47e6T^{2} \) |
| 23 | \( 1 + (1.18e3 + 2.05e3i)T + (-3.21e6 + 5.57e6i)T^{2} \) |
| 29 | \( 1 + (-3.94e3 - 2.27e3i)T + (1.02e7 + 1.77e7i)T^{2} \) |
| 31 | \( 1 + (8.61e3 - 4.97e3i)T + (1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 + 5.94e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (1.05e4 - 6.08e3i)T + (5.79e7 - 1.00e8i)T^{2} \) |
| 43 | \( 1 + (1.51e4 + 8.75e3i)T + (7.35e7 + 1.27e8i)T^{2} \) |
| 47 | \( 1 + (9.29e3 - 1.60e4i)T + (-1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 + 3.27e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + (1.80e3 + 3.13e3i)T + (-3.57e8 + 6.19e8i)T^{2} \) |
| 61 | \( 1 + (-7.91e3 + 1.37e4i)T + (-4.22e8 - 7.31e8i)T^{2} \) |
| 67 | \( 1 + (-4.16e4 + 2.40e4i)T + (6.75e8 - 1.16e9i)T^{2} \) |
| 71 | \( 1 + 3.21e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.47e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + (7.35e4 + 4.24e4i)T + (1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + (-7.54e3 + 1.30e4i)T + (-1.96e9 - 3.41e9i)T^{2} \) |
| 89 | \( 1 - 4.01e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 + (-4.29e3 + 7.44e3i)T + (-4.29e9 - 7.43e9i)T^{2} \) |
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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
−12.69897503573927533599820177536, −11.87918410375606584966399541122, −11.12065261892357467545139210859, −10.27372580292273282195407802211, −8.628131977101305609889102909678, −8.036211023784983215924541982707, −6.96960971240564037408617397602, −4.88138539004669777341740997255, −3.31136873300427358056582404571, −1.86424317551132991588023538621,
0.15150343251816962305487086763, 1.55442340073928014229475512220, 4.21275072978992170742789808197, 5.30062155280928614675933150790, 7.17053628802059797272071133239, 7.909762898216812448149112294669, 8.689811080523842960394400251220, 10.08978560382460015654900254026, 11.31532108158818671411213724071, 11.82373234145340375323715909324
