| L(s) = 1 | + (−5.44 − 6.28i)2-s + (7.57 + 4.86i)3-s + (−5.28 + 36.7i)4-s + (−18.8 + 41.2i)5-s + (−10.6 − 74.0i)6-s + (−66.5 − 19.5i)7-s + (35.7 − 22.9i)8-s + (33.6 + 73.6i)9-s + (361. − 106. i)10-s + (351. − 405. i)11-s + (−218. + 252. i)12-s + (695. − 204. i)13-s + (239. + 524. i)14-s + (−343. + 220. i)15-s + (800. + 235. i)16-s + (−304. − 2.11e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.962 − 1.11i)2-s + (0.485 + 0.312i)3-s + (−0.165 + 1.14i)4-s + (−0.336 + 0.737i)5-s + (−0.120 − 0.839i)6-s + (−0.513 − 0.150i)7-s + (0.197 − 0.126i)8-s + (0.138 + 0.303i)9-s + (1.14 − 0.335i)10-s + (0.875 − 1.01i)11-s + (−0.438 + 0.505i)12-s + (1.14 − 0.335i)13-s + (0.326 + 0.714i)14-s + (−0.393 + 0.252i)15-s + (0.781 + 0.229i)16-s + (−0.255 − 1.77i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.239 + 0.970i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.239 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.628674 - 0.802886i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.628674 - 0.802886i\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + (1.72e3 + 1.86e3i)T \) |
| good | 2 | \( 1 + (5.44 + 6.28i)T + (-4.55 + 31.6i)T^{2} \) |
| 5 | \( 1 + (18.8 - 41.2i)T + (-2.04e3 - 2.36e3i)T^{2} \) |
| 7 | \( 1 + (66.5 + 19.5i)T + (1.41e4 + 9.08e3i)T^{2} \) |
| 11 | \( 1 + (-351. + 405. i)T + (-2.29e4 - 1.59e5i)T^{2} \) |
| 13 | \( 1 + (-695. + 204. i)T + (3.12e5 - 2.00e5i)T^{2} \) |
| 17 | \( 1 + (304. + 2.11e3i)T + (-1.36e6 + 4.00e5i)T^{2} \) |
| 19 | \( 1 + (98.7 - 686. i)T + (-2.37e6 - 6.97e5i)T^{2} \) |
| 29 | \( 1 + (417. + 2.90e3i)T + (-1.96e7 + 5.77e6i)T^{2} \) |
| 31 | \( 1 + (-7.01e3 + 4.50e3i)T + (1.18e7 - 2.60e7i)T^{2} \) |
| 37 | \( 1 + (-57.7 - 126. i)T + (-4.54e7 + 5.24e7i)T^{2} \) |
| 41 | \( 1 + (2.71e3 - 5.94e3i)T + (-7.58e7 - 8.75e7i)T^{2} \) |
| 43 | \( 1 + (-9.01e3 - 5.79e3i)T + (6.10e7 + 1.33e8i)T^{2} \) |
| 47 | \( 1 - 8.48e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + (-3.59e3 - 1.05e3i)T + (3.51e8 + 2.26e8i)T^{2} \) |
| 59 | \( 1 + (-2.88e4 + 8.47e3i)T + (6.01e8 - 3.86e8i)T^{2} \) |
| 61 | \( 1 + (-2.98e4 + 1.91e4i)T + (3.50e8 - 7.68e8i)T^{2} \) |
| 67 | \( 1 + (3.15e4 + 3.63e4i)T + (-1.92e8 + 1.33e9i)T^{2} \) |
| 71 | \( 1 + (1.63e4 + 1.88e4i)T + (-2.56e8 + 1.78e9i)T^{2} \) |
| 73 | \( 1 + (-2.58e3 + 1.80e4i)T + (-1.98e9 - 5.84e8i)T^{2} \) |
| 79 | \( 1 + (3.46e4 - 1.01e4i)T + (2.58e9 - 1.66e9i)T^{2} \) |
| 83 | \( 1 + (-2.49e4 - 5.45e4i)T + (-2.57e9 + 2.97e9i)T^{2} \) |
| 89 | \( 1 + (1.04e4 + 6.74e3i)T + (2.31e9 + 5.07e9i)T^{2} \) |
| 97 | \( 1 + (-1.65e4 + 3.62e4i)T + (-5.62e9 - 6.48e9i)T^{2} \) |
| 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
−13.39126758789456030249294840615, −11.75870311723991345900765041172, −11.09422997339275385868953134419, −10.03165876904086979609304612157, −9.074811539674798912382743733376, −8.031087215084227070202982762442, −6.33408588276808332769840486239, −3.71562840084999080273569593016, −2.74639188489219245777539066274, −0.69678546459359170438092273280,
1.30225020623010757390761942053, 3.98042209089632431734888580527, 6.10954288104607805863700318322, 7.06933887779246937424679672605, 8.462101959652776592947995713396, 8.902679894509097904185457151171, 10.15025280946559807737156916191, 12.06888843646634645342449709245, 12.99114741721067204788738521197, 14.41967625891201103881841341081
