| L(s) = 1 | + (2.35 − 3.23i)2-s + (−1.61 + 0.523i)3-s + (−4.94 − 15.2i)4-s + (−47.4 − 29.6i)5-s + (−2.09 + 6.44i)6-s + 48.2i·7-s + (−60.8 − 19.7i)8-s + (−194. + 141. i)9-s + (−207. + 83.7i)10-s + (−272. − 198. i)11-s + (15.9 + 21.9i)12-s + (−83.4 − 114. i)13-s + (156. + 113. i)14-s + (91.9 + 22.9i)15-s + (−207. + 150. i)16-s + (−1.11e3 − 361. i)17-s + ⋯ |
| L(s) = 1 | + (0.415 − 0.572i)2-s + (−0.103 + 0.0336i)3-s + (−0.154 − 0.475i)4-s + (−0.847 − 0.530i)5-s + (−0.0237 + 0.0731i)6-s + 0.372i·7-s + (−0.336 − 0.109i)8-s + (−0.799 + 0.580i)9-s + (−0.655 + 0.264i)10-s + (−0.679 − 0.493i)11-s + (0.0319 + 0.0439i)12-s + (−0.136 − 0.188i)13-s + (0.213 + 0.154i)14-s + (0.105 + 0.0263i)15-s + (−0.202 + 0.146i)16-s + (−0.934 − 0.303i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.955 - 0.295i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.955 - 0.295i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.0704048 + 0.465215i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0704048 + 0.465215i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-2.35 + 3.23i)T \) |
| 5 | \( 1 + (47.4 + 29.6i)T \) |
| good | 3 | \( 1 + (1.61 - 0.523i)T + (196. - 142. i)T^{2} \) |
| 7 | \( 1 - 48.2iT - 1.68e4T^{2} \) |
| 11 | \( 1 + (272. + 198. i)T + (4.97e4 + 1.53e5i)T^{2} \) |
| 13 | \( 1 + (83.4 + 114. i)T + (-1.14e5 + 3.53e5i)T^{2} \) |
| 17 | \( 1 + (1.11e3 + 361. i)T + (1.14e6 + 8.34e5i)T^{2} \) |
| 19 | \( 1 + (199. - 612. i)T + (-2.00e6 - 1.45e6i)T^{2} \) |
| 23 | \( 1 + (-1.15e3 + 1.58e3i)T + (-1.98e6 - 6.12e6i)T^{2} \) |
| 29 | \( 1 + (885. + 2.72e3i)T + (-1.65e7 + 1.20e7i)T^{2} \) |
| 31 | \( 1 + (-2.48e3 + 7.64e3i)T + (-2.31e7 - 1.68e7i)T^{2} \) |
| 37 | \( 1 + (1.59e3 + 2.19e3i)T + (-2.14e7 + 6.59e7i)T^{2} \) |
| 41 | \( 1 + (-1.21e4 + 8.85e3i)T + (3.58e7 - 1.10e8i)T^{2} \) |
| 43 | \( 1 - 2.66e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + (9.63e3 - 3.12e3i)T + (1.85e8 - 1.34e8i)T^{2} \) |
| 53 | \( 1 + (1.93e4 - 6.29e3i)T + (3.38e8 - 2.45e8i)T^{2} \) |
| 59 | \( 1 + (3.36e4 - 2.44e4i)T + (2.20e8 - 6.79e8i)T^{2} \) |
| 61 | \( 1 + (5.01e3 + 3.64e3i)T + (2.60e8 + 8.03e8i)T^{2} \) |
| 67 | \( 1 + (4.14e4 + 1.34e4i)T + (1.09e9 + 7.93e8i)T^{2} \) |
| 71 | \( 1 + (6.80e3 + 2.09e4i)T + (-1.45e9 + 1.06e9i)T^{2} \) |
| 73 | \( 1 + (3.30e4 - 4.54e4i)T + (-6.40e8 - 1.97e9i)T^{2} \) |
| 79 | \( 1 + (-2.67e4 - 8.23e4i)T + (-2.48e9 + 1.80e9i)T^{2} \) |
| 83 | \( 1 + (9.08e4 + 2.95e4i)T + (3.18e9 + 2.31e9i)T^{2} \) |
| 89 | \( 1 + (-7.98e4 - 5.79e4i)T + (1.72e9 + 5.31e9i)T^{2} \) |
| 97 | \( 1 + (-5.85e4 + 1.90e4i)T + (6.94e9 - 5.04e9i)T^{2} \) |
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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
−13.70817839967022194836141273441, −12.68696108505292599548533501594, −11.56625385055861063259302554817, −10.76208787488758592102021012875, −9.011870996362636566547848250439, −7.88315075702011763080341869660, −5.76525008276881008092222940939, −4.47054497802174176583554786572, −2.66658121630143693687736297885, −0.20246624214651771449074400464,
3.14711576838175932466247378690, 4.71191999508435200619267291596, 6.47673201538214960773881107118, 7.54137582117282900741627513457, 8.870103652189164804428502263701, 10.70183202458633161833126556468, 11.77549592641552281640725272704, 12.95330990990890655040496269238, 14.27099935700396490518605223083, 15.15946270147290115702357898713
