L(s) = 1 | + (5.61 + 0.669i)2-s − 15.7·3-s + (31.1 + 7.52i)4-s − 23.2·5-s + (−88.2 − 10.5i)6-s − 110. i·7-s + (169. + 63.0i)8-s + 4.08·9-s + (−130. − 15.5i)10-s − 468. i·11-s + (−488. − 118. i)12-s − 546. i·13-s + (73.6 − 618. i)14-s + 365.·15-s + (910. + 467. i)16-s − 170.·17-s + ⋯ |
L(s) = 1 | + (0.992 + 0.118i)2-s − 1.00·3-s + (0.971 + 0.235i)4-s − 0.416·5-s + (−1.00 − 0.119i)6-s − 0.848i·7-s + (0.937 + 0.348i)8-s + 0.0168·9-s + (−0.413 − 0.0492i)10-s − 1.16i·11-s + (−0.980 − 0.237i)12-s − 0.897i·13-s + (0.100 − 0.842i)14-s + 0.419·15-s + (0.889 + 0.456i)16-s − 0.143·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0728 + 0.997i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.0728 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.10767 - 1.19156i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.10767 - 1.19156i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-5.61 - 0.669i)T \) |
| 19 | \( 1 + (257. + 1.55e3i)T \) |
good | 3 | \( 1 + 15.7T + 243T^{2} \) |
| 5 | \( 1 + 23.2T + 3.12e3T^{2} \) |
| 7 | \( 1 + 110. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 468. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + 546. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 170.T + 1.41e6T^{2} \) |
| 23 | \( 1 + 3.43e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 5.80e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 2.39e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 6.13e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 3.81e3iT - 1.15e8T^{2} \) |
| 43 | \( 1 - 1.29e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 1.17e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 867. iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 4.51e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.80e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.94e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.08e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 8.65e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.65e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.01e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 1.02e5iT - 5.58e9T^{2} \) |
| 97 | \( 1 + 1.55e5iT - 8.58e9T^{2} \) |
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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
−13.21410096049412949076965452144, −12.16833080701488144926680438447, −11.06671346336774183193845985059, −10.64700981668108041537711787934, −8.301344674319482575443320895351, −6.93392305433389430704987418051, −5.85761372930443328387696541714, −4.67967394504300711813959400331, −3.19930617558687408546463562235, −0.56251850417799994245517987494,
2.01322954597781533771666534228, 4.03681784886748282966161692365, 5.33261951572008441992497608333, 6.28789445408076248952360788076, 7.62684249556390632048646548367, 9.585149195199398028811719340686, 10.99104135081069288547637798209, 11.96921953644022066331145356596, 12.26393787736355265312835450616, 13.74503402738647040169802724254