| L(s) = 1 | + (−2.12 + 5.24i)2-s + (−8.46 + 14.6i)3-s + (−22.9 − 22.2i)4-s + (29.9 − 51.8i)5-s + (−58.9 − 75.5i)6-s + 59.1i·7-s + (165. − 73.0i)8-s + (−21.9 − 38.0i)9-s + (208. + 266. i)10-s − 688. i·11-s + (521. − 148. i)12-s + (858. − 495. i)13-s + (−310. − 125. i)14-s + (506. + 877. i)15-s + (31.1 + 1.02e3i)16-s + (−621. + 1.07e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.375 + 0.926i)2-s + (−0.543 + 0.940i)3-s + (−0.717 − 0.696i)4-s + (0.535 − 0.926i)5-s + (−0.667 − 0.856i)6-s + 0.456i·7-s + (0.914 − 0.403i)8-s + (−0.0902 − 0.156i)9-s + (0.657 + 0.843i)10-s − 1.71i·11-s + (1.04 − 0.297i)12-s + (1.40 − 0.813i)13-s + (−0.422 − 0.171i)14-s + (0.581 + 1.00i)15-s + (0.0304 + 0.999i)16-s + (−0.521 + 0.903i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.597 - 0.801i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.597 - 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.12888 + 0.566530i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.12888 + 0.566530i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (2.12 - 5.24i)T \) |
| 19 | \( 1 + (743. - 1.38e3i)T \) |
| good | 3 | \( 1 + (8.46 - 14.6i)T + (-121.5 - 210. i)T^{2} \) |
| 5 | \( 1 + (-29.9 + 51.8i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 7 | \( 1 - 59.1iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 688. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + (-858. + 495. i)T + (1.85e5 - 3.21e5i)T^{2} \) |
| 17 | \( 1 + (621. - 1.07e3i)T + (-7.09e5 - 1.22e6i)T^{2} \) |
| 23 | \( 1 + (-1.83e3 + 1.06e3i)T + (3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 + (-1.81e3 + 1.04e3i)T + (1.02e7 - 1.77e7i)T^{2} \) |
| 31 | \( 1 - 7.63e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.14e4iT - 6.93e7T^{2} \) |
| 41 | \( 1 + (-7.50e3 - 4.33e3i)T + (5.79e7 + 1.00e8i)T^{2} \) |
| 43 | \( 1 + (-2.26e3 - 1.30e3i)T + (7.35e7 + 1.27e8i)T^{2} \) |
| 47 | \( 1 + (-1.02e4 + 5.92e3i)T + (1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 + (-3.18e4 + 1.84e4i)T + (2.09e8 - 3.62e8i)T^{2} \) |
| 59 | \( 1 + (9.28e3 - 1.60e4i)T + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (-6.22e3 - 1.07e4i)T + (-4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (1.09e4 + 1.90e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 + (-6.06e3 + 1.05e4i)T + (-9.02e8 - 1.56e9i)T^{2} \) |
| 73 | \( 1 + (3.19e4 - 5.53e4i)T + (-1.03e9 - 1.79e9i)T^{2} \) |
| 79 | \( 1 + (2.18e4 - 3.78e4i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 + 7.08e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + (5.71e4 - 3.29e4i)T + (2.79e9 - 4.83e9i)T^{2} \) |
| 97 | \( 1 + (9.61e4 + 5.55e4i)T + (4.29e9 + 7.43e9i)T^{2} \) |
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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
−13.72604267769340091977239441226, −12.94062117097352586409294807099, −11.02251409006367588577141539354, −10.28382958942879804263676829563, −8.816919907676089912811668831672, −8.400607807836956603679916921131, −5.97625628182085975737554186342, −5.60927276004922157729340701608, −4.10434048514357431706619324963, −0.884161737746260882406998854523,
1.13568661620622249226974383078, 2.46753812760431928017865919763, 4.41829318626116449686125069511, 6.60590551983069049156360249467, 7.25719342987215996710108039839, 9.073952023643358421005362474346, 10.22825864106445844556732454016, 11.21714176452454138380031974039, 12.10811733534984288383767615989, 13.27542941918666198707101885828
