| L(s) = 1 | + (−5.24 − 1.61i)2-s + (7.63 − 19.4i)3-s + (−1.52 − 1.04i)4-s + (−57.8 + 8.71i)5-s + (−71.5 + 89.6i)6-s + (−83.4 − 99.2i)7-s + (115. + 145. i)8-s + (−141. − 131. i)9-s + (317. + 47.8i)10-s + (213. − 198. i)11-s + (−31.9 + 21.7i)12-s + (48.9 + 214. i)13-s + (277. + 655. i)14-s + (−271. + 1.19e3i)15-s + (−351. − 894. i)16-s + (73.8 + 984. i)17-s + ⋯ |
| L(s) = 1 | + (−0.927 − 0.286i)2-s + (0.489 − 1.24i)3-s + (−0.0477 − 0.0325i)4-s + (−1.03 + 0.155i)5-s + (−0.811 + 1.01i)6-s + (−0.643 − 0.765i)7-s + (0.640 + 0.802i)8-s + (−0.583 − 0.541i)9-s + (1.00 + 0.151i)10-s + (0.533 − 0.494i)11-s + (−0.0639 + 0.0436i)12-s + (0.0804 + 0.352i)13-s + (0.378 + 0.894i)14-s + (−0.311 + 1.36i)15-s + (−0.343 − 0.873i)16-s + (0.0619 + 0.826i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.168 - 0.985i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.168 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.0690713 + 0.0819098i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0690713 + 0.0819098i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (83.4 + 99.2i)T \) |
| good | 2 | \( 1 + (5.24 + 1.61i)T + (26.4 + 18.0i)T^{2} \) |
| 3 | \( 1 + (-7.63 + 19.4i)T + (-178. - 165. i)T^{2} \) |
| 5 | \( 1 + (57.8 - 8.71i)T + (2.98e3 - 921. i)T^{2} \) |
| 11 | \( 1 + (-213. + 198. i)T + (1.20e4 - 1.60e5i)T^{2} \) |
| 13 | \( 1 + (-48.9 - 214. i)T + (-3.34e5 + 1.61e5i)T^{2} \) |
| 17 | \( 1 + (-73.8 - 984. i)T + (-1.40e6 + 2.11e5i)T^{2} \) |
| 19 | \( 1 + (1.10e3 - 1.90e3i)T + (-1.23e6 - 2.14e6i)T^{2} \) |
| 23 | \( 1 + (-62.6 + 835. i)T + (-6.36e6 - 9.59e5i)T^{2} \) |
| 29 | \( 1 + (1.99e3 + 959. i)T + (1.27e7 + 1.60e7i)T^{2} \) |
| 31 | \( 1 + (895. + 1.55e3i)T + (-1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 + (3.86e3 - 2.63e3i)T + (2.53e7 - 6.45e7i)T^{2} \) |
| 41 | \( 1 + (1.06e4 + 1.33e4i)T + (-2.57e7 + 1.12e8i)T^{2} \) |
| 43 | \( 1 + (1.24e4 - 1.55e4i)T + (-3.27e7 - 1.43e8i)T^{2} \) |
| 47 | \( 1 + (-222. - 68.7i)T + (1.89e8 + 1.29e8i)T^{2} \) |
| 53 | \( 1 + (2.92e4 + 1.99e4i)T + (1.52e8 + 3.89e8i)T^{2} \) |
| 59 | \( 1 + (-2.78e4 - 4.20e3i)T + (6.83e8 + 2.10e8i)T^{2} \) |
| 61 | \( 1 + (-2.57e4 + 1.75e4i)T + (3.08e8 - 7.86e8i)T^{2} \) |
| 67 | \( 1 + (2.07e4 + 3.59e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 + (-345. + 166. i)T + (1.12e9 - 1.41e9i)T^{2} \) |
| 73 | \( 1 + (5.81e4 - 1.79e4i)T + (1.71e9 - 1.16e9i)T^{2} \) |
| 79 | \( 1 + (1.46e4 - 2.54e4i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 + (2.23e4 - 9.79e4i)T + (-3.54e9 - 1.70e9i)T^{2} \) |
| 89 | \( 1 + (-7.53e4 - 6.99e4i)T + (4.17e8 + 5.56e9i)T^{2} \) |
| 97 | \( 1 + 1.11e5T + 8.58e9T^{2} \) |
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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
−13.70305368844678493876208367102, −12.60126226302812207110940501471, −11.33034753345966011436371916524, −10.08602307054036897095888486165, −8.524524157232908115895097447941, −7.80465383051761932192982328828, −6.57586800369547669688910396782, −3.79958340589070200224247213937, −1.56948972313563058790293511529, −0.07382847530901417711096627526,
3.41577825818569806995622164295, 4.65930978089802717576632734350, 7.08991411675879814600116400574, 8.578132393253752528470428094668, 9.218810056491186984345689442834, 10.20788466748333684051857167137, 11.73873620788458998606612710892, 13.10388878814152017889365287012, 14.89235144077176833271070503539, 15.68655760335410513105608777591
