| L(s) = 1 | + (3.76 + 4.34i)2-s + (13.1 + 8.42i)3-s + (4.41 − 30.6i)4-s + (172. + 78.7i)5-s + (12.7 + 88.6i)6-s + (43.8 − 149. i)7-s + (459. − 295. i)8-s + (100. + 221. i)9-s + (306. + 1.04e3i)10-s + (−1.27e3 − 1.10e3i)11-s + (316. − 365. i)12-s + (2.55e3 − 749. i)13-s + (813. − 371. i)14-s + (1.59e3 + 2.48e3i)15-s + (1.10e3 + 324. i)16-s + (−4.03e3 + 580. i)17-s + ⋯ |
| L(s) = 1 | + (0.470 + 0.542i)2-s + (0.485 + 0.312i)3-s + (0.0689 − 0.479i)4-s + (1.38 + 0.630i)5-s + (0.0589 + 0.410i)6-s + (0.127 − 0.435i)7-s + (0.896 − 0.576i)8-s + (0.138 + 0.303i)9-s + (0.306 + 1.04i)10-s + (−0.956 − 0.828i)11-s + (0.183 − 0.211i)12-s + (1.16 − 0.341i)13-s + (0.296 − 0.135i)14-s + (0.473 + 0.737i)15-s + (0.269 + 0.0791i)16-s + (−0.821 + 0.118i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.883 - 0.469i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.883 - 0.469i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(3.54952 + 0.884525i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.54952 + 0.884525i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-13.1 - 8.42i)T \) |
| 23 | \( 1 + (9.99e3 - 6.93e3i)T \) |
| good | 2 | \( 1 + (-3.76 - 4.34i)T + (-9.10 + 63.3i)T^{2} \) |
| 5 | \( 1 + (-172. - 78.7i)T + (1.02e4 + 1.18e4i)T^{2} \) |
| 7 | \( 1 + (-43.8 + 149. i)T + (-9.89e4 - 6.36e4i)T^{2} \) |
| 11 | \( 1 + (1.27e3 + 1.10e3i)T + (2.52e5 + 1.75e6i)T^{2} \) |
| 13 | \( 1 + (-2.55e3 + 749. i)T + (4.06e6 - 2.60e6i)T^{2} \) |
| 17 | \( 1 + (4.03e3 - 580. i)T + (2.31e7 - 6.80e6i)T^{2} \) |
| 19 | \( 1 + (-1.15e4 - 1.66e3i)T + (4.51e7 + 1.32e7i)T^{2} \) |
| 29 | \( 1 + (-6.02e3 - 4.18e4i)T + (-5.70e8 + 1.67e8i)T^{2} \) |
| 31 | \( 1 + (2.57e4 - 1.65e4i)T + (3.68e8 - 8.07e8i)T^{2} \) |
| 37 | \( 1 + (-263. + 120. i)T + (1.68e9 - 1.93e9i)T^{2} \) |
| 41 | \( 1 + (-3.03e4 + 6.63e4i)T + (-3.11e9 - 3.58e9i)T^{2} \) |
| 43 | \( 1 + (-2.58e3 + 4.01e3i)T + (-2.62e9 - 5.75e9i)T^{2} \) |
| 47 | \( 1 + 7.85e4T + 1.07e10T^{2} \) |
| 53 | \( 1 + (-3.18e4 + 1.08e5i)T + (-1.86e10 - 1.19e10i)T^{2} \) |
| 59 | \( 1 + (3.02e5 - 8.88e4i)T + (3.54e10 - 2.28e10i)T^{2} \) |
| 61 | \( 1 + (1.47e5 + 2.28e5i)T + (-2.14e10 + 4.68e10i)T^{2} \) |
| 67 | \( 1 + (3.71e5 - 3.21e5i)T + (1.28e10 - 8.95e10i)T^{2} \) |
| 71 | \( 1 + (3.47e5 + 4.00e5i)T + (-1.82e10 + 1.26e11i)T^{2} \) |
| 73 | \( 1 + (-1.82e4 + 1.27e5i)T + (-1.45e11 - 4.26e10i)T^{2} \) |
| 79 | \( 1 + (6.65e4 + 2.26e5i)T + (-2.04e11 + 1.31e11i)T^{2} \) |
| 83 | \( 1 + (-9.08e4 + 4.14e4i)T + (2.14e11 - 2.47e11i)T^{2} \) |
| 89 | \( 1 + (1.56e5 - 2.43e5i)T + (-2.06e11 - 4.52e11i)T^{2} \) |
| 97 | \( 1 + (6.57e5 + 3.00e5i)T + (5.45e11 + 6.29e11i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.77791556434379254969290486635, −13.28432351640073495559564015803, −10.82889236393439449027169092996, −10.37826989221298690917312036235, −9.112228034310297079384860148854, −7.45708680597037666974737006960, −6.09461519592264303591415632135, −5.26854289729608991484704607571, −3.30524334753726927290756797194, −1.53157086932067476674575452142,
1.69674785386644212928102374613, 2.64633492003728223822393976661, 4.52326099984363573845782181360, 5.91246574249295392619523819516, 7.66086049911307414166134106899, 8.872962368836240433350212963754, 9.955280070627597738646783889747, 11.45925206373035129335563039853, 12.60940756423424653257739051463, 13.44743579688208462959070052576
