L(s) = 1 | + (−1.41 − 0.0853i)2-s + (−0.290 − 1.92i)3-s + (1.98 + 0.240i)4-s + (−1.10 − 0.434i)5-s + (0.245 + 2.74i)6-s + (1.77 + 1.96i)7-s + (−2.78 − 0.509i)8-s + (−0.752 + 0.232i)9-s + (1.52 + 0.708i)10-s + (1.15 − 3.74i)11-s + (−0.112 − 3.89i)12-s + (4.65 − 1.06i)13-s + (−2.33 − 2.92i)14-s + (−0.515 + 2.25i)15-s + (3.88 + 0.956i)16-s + (2.11 − 1.44i)17-s + ⋯ |
L(s) = 1 | + (−0.998 − 0.0603i)2-s + (−0.167 − 1.11i)3-s + (0.992 + 0.120i)4-s + (−0.495 − 0.194i)5-s + (0.100 + 1.11i)6-s + (0.671 + 0.741i)7-s + (−0.983 − 0.180i)8-s + (−0.250 + 0.0773i)9-s + (0.482 + 0.224i)10-s + (0.348 − 1.12i)11-s + (−0.0324 − 1.12i)12-s + (1.29 − 0.294i)13-s + (−0.625 − 0.780i)14-s + (−0.133 + 0.583i)15-s + (0.970 + 0.239i)16-s + (0.512 − 0.349i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.235 + 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.235 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.513719 - 0.653282i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.513719 - 0.653282i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.41 + 0.0853i)T \) |
| 7 | \( 1 + (-1.77 - 1.96i)T \) |
good | 3 | \( 1 + (0.290 + 1.92i)T + (-2.86 + 0.884i)T^{2} \) |
| 5 | \( 1 + (1.10 + 0.434i)T + (3.66 + 3.40i)T^{2} \) |
| 11 | \( 1 + (-1.15 + 3.74i)T + (-9.08 - 6.19i)T^{2} \) |
| 13 | \( 1 + (-4.65 + 1.06i)T + (11.7 - 5.64i)T^{2} \) |
| 17 | \( 1 + (-2.11 + 1.44i)T + (6.21 - 15.8i)T^{2} \) |
| 19 | \( 1 + (3.04 - 1.75i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3.64 + 2.48i)T + (8.40 + 21.4i)T^{2} \) |
| 29 | \( 1 + (2.00 - 4.16i)T + (-18.0 - 22.6i)T^{2} \) |
| 31 | \( 1 + (-1.49 + 2.59i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.67 + 0.125i)T + (36.5 - 5.51i)T^{2} \) |
| 41 | \( 1 + (-1.46 - 1.83i)T + (-9.12 + 39.9i)T^{2} \) |
| 43 | \( 1 + (6.23 + 4.97i)T + (9.56 + 41.9i)T^{2} \) |
| 47 | \( 1 + (-8.15 + 7.56i)T + (3.51 - 46.8i)T^{2} \) |
| 53 | \( 1 + (0.517 + 0.0387i)T + (52.4 + 7.89i)T^{2} \) |
| 59 | \( 1 + (-9.32 + 3.66i)T + (43.2 - 40.1i)T^{2} \) |
| 61 | \( 1 + (-3.74 + 0.280i)T + (60.3 - 9.09i)T^{2} \) |
| 67 | \( 1 + (9.58 + 5.53i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (7.89 - 3.80i)T + (44.2 - 55.5i)T^{2} \) |
| 73 | \( 1 + (-1.56 - 1.45i)T + (5.45 + 72.7i)T^{2} \) |
| 79 | \( 1 + (-6.40 - 11.0i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (5.80 + 1.32i)T + (74.7 + 36.0i)T^{2} \) |
| 89 | \( 1 + (-7.80 + 2.40i)T + (73.5 - 50.1i)T^{2} \) |
| 97 | \( 1 - 12.3T + 97T^{2} \) |
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
−11.24007575988053538656498803575, −10.20295931842682347995273616081, −8.725348933938717494380973716244, −8.367933397281360952068023689167, −7.58304438542182465370951827725, −6.36783047083279749070499121871, −5.76833127595901689768800577245, −3.69480726296304295256799738379, −2.09043725860805543391642801635, −0.845643593017996255347525995849,
1.62980535992684642746428837650, 3.69110575979726194355039256363, 4.44948593239185697962254903065, 5.94308048184311351541772585088, 7.16603329351490071374734564978, 7.918292713520362415241548557984, 8.962675339410353092937393950673, 9.882463311915726473628250098274, 10.51619529339941542668735872903, 11.24230208606101282224966947519