| L(s) = 1 | + (−1.44 − 1.38i)2-s + (0.187 + 3.99i)4-s − 5.24·5-s + (−6.31 + 3.01i)7-s + (5.24 − 6.03i)8-s + (7.58 + 7.24i)10-s + 3.67i·11-s + 12.5·13-s + (13.3 + 4.36i)14-s + (−15.9 + 1.49i)16-s + 18.2i·17-s − 4.94·19-s + (−0.982 − 20.9i)20-s + (5.08 − 5.32i)22-s − 1.93·23-s + ⋯ |
| L(s) = 1 | + (−0.723 − 0.690i)2-s + (0.0468 + 0.998i)4-s − 1.04·5-s + (−0.902 + 0.430i)7-s + (0.655 − 0.754i)8-s + (0.758 + 0.724i)10-s + 0.334i·11-s + 0.962·13-s + (0.950 + 0.312i)14-s + (−0.995 + 0.0935i)16-s + 1.07i·17-s − 0.260·19-s + (−0.0491 − 1.04i)20-s + (0.230 − 0.242i)22-s − 0.0841·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.267 + 0.963i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.267 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5327048186\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5327048186\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.44 + 1.38i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (6.31 - 3.01i)T \) |
| good | 5 | \( 1 + 5.24T + 25T^{2} \) |
| 11 | \( 1 - 3.67iT - 121T^{2} \) |
| 13 | \( 1 - 12.5T + 169T^{2} \) |
| 17 | \( 1 - 18.2iT - 289T^{2} \) |
| 19 | \( 1 + 4.94T + 361T^{2} \) |
| 23 | \( 1 + 1.93T + 529T^{2} \) |
| 29 | \( 1 + 41.0iT - 841T^{2} \) |
| 31 | \( 1 + 22.7iT - 961T^{2} \) |
| 37 | \( 1 + 46.7iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 66.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 41.9iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 53.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 61.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 70.9T + 3.48e3T^{2} \) |
| 61 | \( 1 - 82.0T + 3.72e3T^{2} \) |
| 67 | \( 1 + 1.63iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 10.0T + 5.04e3T^{2} \) |
| 73 | \( 1 + 64.4iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 149.T + 6.24e3T^{2} \) |
| 83 | \( 1 - 12.9T + 6.88e3T^{2} \) |
| 89 | \( 1 + 130. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 103. iT - 9.40e3T^{2} \) |
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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
−10.39873858708694141897436758808, −9.656704752882994880831723061894, −8.611019733351347240973790097052, −8.064464422882259546264836175476, −7.00014072518156804291202155531, −5.97115378912683404888673437903, −4.11443612988438430088072922489, −3.56750808141035824848293688112, −2.15932378157688478445979387171, −0.35554562008357314675347041227,
0.965423547697047115895809548954, 3.12225931384832708049125259206, 4.29146692062202045881674997837, 5.58722971065288696790807181876, 6.69818367641588040625125806020, 7.28491182402730255383974293606, 8.311343660341812716537619483400, 8.994740898737054442701290179919, 9.990672590016435830970858632152, 10.86443568546000989285503454430
