| L(s) = 1 | + (2.18 + 2.52i)2-s + (7.57 + 4.86i)3-s + (2.96 − 20.6i)4-s + (−1.08 + 2.38i)5-s + (4.28 + 29.7i)6-s + (60.1 + 17.6i)7-s + (148. − 95.4i)8-s + (33.6 + 73.6i)9-s + (−8.40 + 2.46i)10-s + (66.0 − 76.2i)11-s + (122. − 141. i)12-s + (980. − 287. i)13-s + (86.9 + 190. i)14-s + (−19.8 + 12.7i)15-s + (−73.4 − 21.5i)16-s + (102. + 714. i)17-s + ⋯ |
| L(s) = 1 | + (0.386 + 0.446i)2-s + (0.485 + 0.312i)3-s + (0.0926 − 0.644i)4-s + (−0.0194 + 0.0426i)5-s + (0.0485 + 0.337i)6-s + (0.463 + 0.136i)7-s + (0.820 − 0.527i)8-s + (0.138 + 0.303i)9-s + (−0.0265 + 0.00780i)10-s + (0.164 − 0.189i)11-s + (0.246 − 0.284i)12-s + (1.60 − 0.472i)13-s + (0.118 + 0.259i)14-s + (−0.0227 + 0.0146i)15-s + (−0.0717 − 0.0210i)16-s + (0.0862 + 0.600i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 - 0.246i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.969 - 0.246i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.83463 + 0.355321i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.83463 + 0.355321i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-7.57 - 4.86i)T \) |
| 23 | \( 1 + (830. - 2.39e3i)T \) |
| good | 2 | \( 1 + (-2.18 - 2.52i)T + (-4.55 + 31.6i)T^{2} \) |
| 5 | \( 1 + (1.08 - 2.38i)T + (-2.04e3 - 2.36e3i)T^{2} \) |
| 7 | \( 1 + (-60.1 - 17.6i)T + (1.41e4 + 9.08e3i)T^{2} \) |
| 11 | \( 1 + (-66.0 + 76.2i)T + (-2.29e4 - 1.59e5i)T^{2} \) |
| 13 | \( 1 + (-980. + 287. i)T + (3.12e5 - 2.00e5i)T^{2} \) |
| 17 | \( 1 + (-102. - 714. i)T + (-1.36e6 + 4.00e5i)T^{2} \) |
| 19 | \( 1 + (-118. + 825. i)T + (-2.37e6 - 6.97e5i)T^{2} \) |
| 29 | \( 1 + (366. + 2.54e3i)T + (-1.96e7 + 5.77e6i)T^{2} \) |
| 31 | \( 1 + (-332. + 213. i)T + (1.18e7 - 2.60e7i)T^{2} \) |
| 37 | \( 1 + (2.44e3 + 5.36e3i)T + (-4.54e7 + 5.24e7i)T^{2} \) |
| 41 | \( 1 + (5.06e3 - 1.10e4i)T + (-7.58e7 - 8.75e7i)T^{2} \) |
| 43 | \( 1 + (1.41e4 + 9.08e3i)T + (6.10e7 + 1.33e8i)T^{2} \) |
| 47 | \( 1 - 6.57e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + (1.61e4 + 4.74e3i)T + (3.51e8 + 2.26e8i)T^{2} \) |
| 59 | \( 1 + (1.20e4 - 3.54e3i)T + (6.01e8 - 3.86e8i)T^{2} \) |
| 61 | \( 1 + (2.96e4 - 1.90e4i)T + (3.50e8 - 7.68e8i)T^{2} \) |
| 67 | \( 1 + (2.29e4 + 2.64e4i)T + (-1.92e8 + 1.33e9i)T^{2} \) |
| 71 | \( 1 + (-3.15e4 - 3.64e4i)T + (-2.56e8 + 1.78e9i)T^{2} \) |
| 73 | \( 1 + (-5.65e3 + 3.93e4i)T + (-1.98e9 - 5.84e8i)T^{2} \) |
| 79 | \( 1 + (7.87e3 - 2.31e3i)T + (2.58e9 - 1.66e9i)T^{2} \) |
| 83 | \( 1 + (-2.99e4 - 6.56e4i)T + (-2.57e9 + 2.97e9i)T^{2} \) |
| 89 | \( 1 + (-9.74e4 - 6.26e4i)T + (2.31e9 + 5.07e9i)T^{2} \) |
| 97 | \( 1 + (4.46e4 - 9.78e4i)T + (-5.62e9 - 6.48e9i)T^{2} \) |
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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
−13.87205843569562823070318468926, −13.19486752915214860434797467810, −11.34470763975639731788516999519, −10.46932691070772531376485599254, −9.128079695907089184618496392231, −7.908524444351824649501031266378, −6.36509846786474819025583439368, −5.18308033217114452512248997138, −3.66005175358189037132534393872, −1.44908974169751481687771073347,
1.62456988338209847091622428063, 3.25460954326704947238218534609, 4.53218058363659401665038674985, 6.59336470096628825213040711869, 7.972397926335693407079913134976, 8.826986399025371194098148732366, 10.60270855674648323575730096214, 11.67640106260996470543269916657, 12.61235335233690060060953047261, 13.68692645990689056605487538078
