L(s) = 1 | + (−7.82 + 13.5i)5-s + (−13.7 + 12.4i)7-s + (−34.2 + 19.7i)11-s + (55.5 + 32.0i)13-s − 56.6·17-s + 117. i·19-s + (6.59 + 3.80i)23-s + (−59.9 − 103. i)25-s + (−39.8 + 22.9i)29-s + (251. + 145. i)31-s + (−60.6 − 283. i)35-s − 335.·37-s + (−97.2 + 168. i)41-s + (152. + 263. i)43-s + (−318. − 550. i)47-s + ⋯ |
L(s) = 1 | + (−0.699 + 1.21i)5-s + (−0.742 + 0.670i)7-s + (−0.937 + 0.541i)11-s + (1.18 + 0.684i)13-s − 0.808·17-s + 1.41i·19-s + (0.0597 + 0.0345i)23-s + (−0.479 − 0.830i)25-s + (−0.254 + 0.147i)29-s + (1.45 + 0.842i)31-s + (−0.293 − 1.36i)35-s − 1.49·37-s + (−0.370 + 0.641i)41-s + (0.540 + 0.935i)43-s + (−0.987 − 1.70i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.600 + 0.799i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.600 + 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.5677977023\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5677977023\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (13.7 - 12.4i)T \) |
good | 5 | \( 1 + (7.82 - 13.5i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (34.2 - 19.7i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-55.5 - 32.0i)T + (1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + 56.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 117. iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (-6.59 - 3.80i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (39.8 - 22.9i)T + (1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-251. - 145. i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + 335.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (97.2 - 168. i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-152. - 263. i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (318. + 550. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + 274. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (-258. + 448. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-142. + 82.1i)T + (1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-368. + 637. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 599. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 214. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + (454. + 786. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-389. - 675. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 - 443.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-337. + 194. i)T + (4.56e5 - 7.90e5i)T^{2} \) |
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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
−10.46062192860396395440668560055, −9.859886748116754630117642521995, −8.648211423280024527479425543100, −7.987882961056927478126360356558, −6.77020058542970026459503413608, −6.45494058066688508119437017806, −5.18946908494086666115939956093, −3.82498317104767017635153464118, −3.11630847721928125272176875376, −1.96138647483962563010700988346,
0.19500713610379116321286214375, 0.907259096284257041663101757373, 2.80961448033111888952289307636, 3.89679356332762602453165030788, 4.74334371041953288884944901893, 5.75455916721503532134050673027, 6.80434762371397653228604340804, 7.83846366827575245884281222053, 8.555115971180049667723125171153, 9.180968013692065940799788554435