| L(s) = 1 | + (13.5 − 7.79i)3-s + (201. + 116. i)5-s + (−184. − 289. i)7-s + (121.5 − 210. i)9-s + (460. + 796. i)11-s − 536. i·13-s + 3.62e3·15-s + (2.61e3 − 1.50e3i)17-s + (−7.37e3 − 4.25e3i)19-s + (−4.74e3 − 2.46e3i)21-s + (1.05e4 − 1.82e4i)23-s + (1.92e4 + 3.33e4i)25-s − 3.78e3i·27-s − 3.91e3·29-s + (1.99e4 − 1.15e4i)31-s + ⋯ |
| L(s) = 1 | + (0.5 − 0.288i)3-s + (1.61 + 0.930i)5-s + (−0.537 − 0.843i)7-s + (0.166 − 0.288i)9-s + (0.345 + 0.598i)11-s − 0.244i·13-s + 1.07·15-s + (0.531 − 0.306i)17-s + (−1.07 − 0.620i)19-s + (−0.512 − 0.266i)21-s + (0.865 − 1.49i)23-s + (1.23 + 2.13i)25-s − 0.192i·27-s − 0.160·29-s + (0.669 − 0.386i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.821 + 0.569i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.821 + 0.569i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(3.580846333\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.580846333\) |
| \(L(4)\) |
|
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 | \( 1 + (184. + 289. i)T \) |
| good | 5 | \( 1 + (-201. - 116. i)T + (7.81e3 + 1.35e4i)T^{2} \) |
| 11 | \( 1 + (-460. - 796. i)T + (-8.85e5 + 1.53e6i)T^{2} \) |
| 13 | \( 1 + 536. iT - 4.82e6T^{2} \) |
| 17 | \( 1 + (-2.61e3 + 1.50e3i)T + (1.20e7 - 2.09e7i)T^{2} \) |
| 19 | \( 1 + (7.37e3 + 4.25e3i)T + (2.35e7 + 4.07e7i)T^{2} \) |
| 23 | \( 1 + (-1.05e4 + 1.82e4i)T + (-7.40e7 - 1.28e8i)T^{2} \) |
| 29 | \( 1 + 3.91e3T + 5.94e8T^{2} \) |
| 31 | \( 1 + (-1.99e4 + 1.15e4i)T + (4.43e8 - 7.68e8i)T^{2} \) |
| 37 | \( 1 + (-2.20e4 + 3.82e4i)T + (-1.28e9 - 2.22e9i)T^{2} \) |
| 41 | \( 1 + 5.66e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 1.20e5T + 6.32e9T^{2} \) |
| 47 | \( 1 + (1.31e5 + 7.58e4i)T + (5.38e9 + 9.33e9i)T^{2} \) |
| 53 | \( 1 + (-1.15e5 - 2.00e5i)T + (-1.10e10 + 1.91e10i)T^{2} \) |
| 59 | \( 1 + (-7.46e3 + 4.30e3i)T + (2.10e10 - 3.65e10i)T^{2} \) |
| 61 | \( 1 + (-6.85e4 - 3.95e4i)T + (2.57e10 + 4.46e10i)T^{2} \) |
| 67 | \( 1 + (-1.56e5 - 2.71e5i)T + (-4.52e10 + 7.83e10i)T^{2} \) |
| 71 | \( 1 - 2.12e5T + 1.28e11T^{2} \) |
| 73 | \( 1 + (1.98e5 - 1.14e5i)T + (7.56e10 - 1.31e11i)T^{2} \) |
| 79 | \( 1 + (3.79e5 - 6.56e5i)T + (-1.21e11 - 2.10e11i)T^{2} \) |
| 83 | \( 1 + 1.61e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + (4.86e5 + 2.80e5i)T + (2.48e11 + 4.30e11i)T^{2} \) |
| 97 | \( 1 + 1.99e5iT - 8.32e11T^{2} \) |
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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
−10.29072684864144793737881429859, −9.669309351659236015145988274592, −8.763647067910604753838945117059, −7.22461548499002216142778516109, −6.74635232800308954414880436066, −5.82781536002357507367819742290, −4.32233530381239334121209983467, −2.91077994180661801034459583059, −2.18996415811791980342764399688, −0.826409653357325263678098614459,
1.19304548048622912135470727506, 2.15424717468343957309587729491, 3.33242808122474221995327403571, 4.82985716386963032389033655512, 5.77771017195591010877036933928, 6.41909300808388260102943518129, 8.179093160524059879038335979432, 8.970929405577895376554801628528, 9.534432817149240130561383804477, 10.24539084417822191837095138705
