| L(s) = 1 | + (0.371 + 1.96i)2-s + (0.824 − 1.42i)3-s + (−3.72 + 1.46i)4-s + (3.95 − 2.28i)5-s + (3.11 + 1.08i)6-s + (−4.25 − 6.77i)8-s + (3.14 + 5.43i)9-s + (5.94 + 6.91i)10-s + (6.18 − 10.7i)11-s + (−0.984 + 6.52i)12-s − 18.3i·13-s − 7.52i·15-s + (11.7 − 10.8i)16-s + (−6.51 + 11.2i)17-s + (−9.52 + 8.19i)18-s + (−1.51 − 2.61i)19-s + ⋯ |
| L(s) = 1 | + (0.185 + 0.982i)2-s + (0.274 − 0.475i)3-s + (−0.930 + 0.365i)4-s + (0.790 − 0.456i)5-s + (0.518 + 0.181i)6-s + (−0.531 − 0.846i)8-s + (0.348 + 0.604i)9-s + (0.594 + 0.691i)10-s + (0.562 − 0.974i)11-s + (−0.0820 + 0.543i)12-s − 1.41i·13-s − 0.501i·15-s + (0.733 − 0.679i)16-s + (−0.383 + 0.663i)17-s + (−0.529 + 0.455i)18-s + (−0.0796 − 0.137i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.162i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.986 - 0.162i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.17050 + 0.178037i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.17050 + 0.178037i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.371 - 1.96i)T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + (-0.824 + 1.42i)T + (-4.5 - 7.79i)T^{2} \) |
| 5 | \( 1 + (-3.95 + 2.28i)T + (12.5 - 21.6i)T^{2} \) |
| 11 | \( 1 + (-6.18 + 10.7i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + 18.3iT - 169T^{2} \) |
| 17 | \( 1 + (6.51 - 11.2i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (1.51 + 2.61i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-26.2 + 15.1i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + 22.7iT - 841T^{2} \) |
| 31 | \( 1 + (-19.5 - 11.2i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (11.9 - 6.88i)T + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 - 60.5T + 1.68e3T^{2} \) |
| 43 | \( 1 - 39.0T + 1.84e3T^{2} \) |
| 47 | \( 1 + (17.6 - 10.1i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (4.12 + 2.38i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-5.86 + 10.1i)T + (-1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (94.3 - 54.4i)T + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-39.5 + 68.5i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 12.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (49.2 - 85.3i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (113. - 65.6i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + 28.3T + 6.88e3T^{2} \) |
| 89 | \( 1 + (78.7 + 136. i)T + (-3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 - 39.6T + 9.40e3T^{2} \) |
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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
−10.99993009542213326449417643964, −10.01213917027666282526103899253, −8.911658017585133041083792305966, −8.306703534011247230170610366653, −7.38751527371292477429892474429, −6.25402325571303348871283797673, −5.54867333040609309773596112592, −4.43677928078292485581045750426, −2.89943258167116643089374700676, −1.02400597019138794760947404753,
1.52009368061662089569858036566, 2.70248540661583144048292182329, 4.00872894005450256655286717158, 4.77952001535617198485230918313, 6.23401656128788844605081586918, 7.16869029569836072715693953122, 9.042183334040886838227631560459, 9.339619984481274902405465117553, 10.05987410511281331241390335079, 11.03767181878098847701902252745
