| L(s) = 1 | + (−2.64 + 4.99i)2-s + (−15.2 + 26.4i)3-s + (−17.9 − 26.4i)4-s + (−23.0 + 39.9i)5-s + (−91.6 − 146. i)6-s − 131. i·7-s + (179. − 19.8i)8-s + (−343. − 594. i)9-s + (−138. − 221. i)10-s + 582. i·11-s + (973. − 71.4i)12-s + (−337. + 195. i)13-s + (655. + 347. i)14-s + (−704. − 1.21e3i)15-s + (−376. + 952. i)16-s + (359. − 623. i)17-s + ⋯ |
| L(s) = 1 | + (−0.467 + 0.883i)2-s + (−0.978 + 1.69i)3-s + (−0.562 − 0.827i)4-s + (−0.413 + 0.715i)5-s + (−1.03 − 1.65i)6-s − 1.01i·7-s + (0.993 − 0.109i)8-s + (−1.41 − 2.44i)9-s + (−0.439 − 0.699i)10-s + 1.45i·11-s + (1.95 − 0.143i)12-s + (−0.554 + 0.320i)13-s + (0.893 + 0.473i)14-s + (−0.808 − 1.39i)15-s + (−0.368 + 0.929i)16-s + (0.301 − 0.522i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0550i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0550i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.103976 - 0.00286223i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.103976 - 0.00286223i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (2.64 - 4.99i)T \) |
| 19 | \( 1 + (1.50e3 - 461. i)T \) |
| good | 3 | \( 1 + (15.2 - 26.4i)T + (-121.5 - 210. i)T^{2} \) |
| 5 | \( 1 + (23.0 - 39.9i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 7 | \( 1 + 131. iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 582. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + (337. - 195. i)T + (1.85e5 - 3.21e5i)T^{2} \) |
| 17 | \( 1 + (-359. + 623. i)T + (-7.09e5 - 1.22e6i)T^{2} \) |
| 23 | \( 1 + (1.69e3 - 981. i)T + (3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 + (-6.02e3 + 3.47e3i)T + (1.02e7 - 1.77e7i)T^{2} \) |
| 31 | \( 1 + 832.T + 2.86e7T^{2} \) |
| 37 | \( 1 + 8.81e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 + (382. + 220. i)T + (5.79e7 + 1.00e8i)T^{2} \) |
| 43 | \( 1 + (3.21e3 + 1.85e3i)T + (7.35e7 + 1.27e8i)T^{2} \) |
| 47 | \( 1 + (-2.41e3 + 1.39e3i)T + (1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 + (1.61e4 - 9.31e3i)T + (2.09e8 - 3.62e8i)T^{2} \) |
| 59 | \( 1 + (1.27e4 - 2.19e4i)T + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (723. + 1.25e3i)T + (-4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (1.51e4 + 2.61e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 + (-3.50e4 + 6.07e4i)T + (-9.02e8 - 1.56e9i)T^{2} \) |
| 73 | \( 1 + (1.42e3 - 2.46e3i)T + (-1.03e9 - 1.79e9i)T^{2} \) |
| 79 | \( 1 + (-2.25e4 + 3.91e4i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 + 2.72e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + (-7.85e4 + 4.53e4i)T + (2.79e9 - 4.83e9i)T^{2} \) |
| 97 | \( 1 + (1.08e5 + 6.27e4i)T + (4.29e9 + 7.43e9i)T^{2} \) |
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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
−14.19532710947579323667611302254, −12.06901240844464134576247144219, −10.74406771131863341382868067495, −10.18619304868295460220190797611, −9.347310408526919889668302317225, −7.46522841497032149367899845378, −6.38112986150837575887294979976, −4.83288856126656169826489998339, −4.04205361416091962956576809063, −0.07358243476545157472549044993,
1.05941867878596893756582929834, 2.56730860712027213994556716102, 5.10933836826450280700607878032, 6.41832380616836892418604399779, 8.172598178592363903831974991246, 8.502184656947561292894952355116, 10.64753008747243989426608736149, 11.65565749211663816333867853024, 12.35382190941060512020405209590, 12.88079960080860284095961359986
