| L(s) = 1 | − 5.84i·2-s + 12.3i·3-s − 2.10·4-s + 72.1·6-s − 135. i·7-s − 174. i·8-s + 90.3·9-s + 564.·11-s − 26.0i·12-s − 169i·13-s − 788.·14-s − 1.08e3·16-s − 1.44e3i·17-s − 527. i·18-s + 530.·19-s + ⋯ |
| L(s) = 1 | − 1.03i·2-s + 0.792i·3-s − 0.0658·4-s + 0.818·6-s − 1.04i·7-s − 0.964i·8-s + 0.371·9-s + 1.40·11-s − 0.0521i·12-s − 0.277i·13-s − 1.07·14-s − 1.06·16-s − 1.21i·17-s − 0.383i·18-s + 0.337·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.565074570\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.565074570\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 13 | \( 1 + 169iT \) |
| good | 2 | \( 1 + 5.84iT - 32T^{2} \) |
| 3 | \( 1 - 12.3iT - 243T^{2} \) |
| 7 | \( 1 + 135. iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 564.T + 1.61e5T^{2} \) |
| 17 | \( 1 + 1.44e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 530.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.71e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 3.32e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.86e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.02e4iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 1.79e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 7.56e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 2.40e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 5.49e3iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 4.54e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.93e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.99e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 4.91e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.13e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 4.84e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.06e5iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 2.33e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 7.75e4iT - 8.58e9T^{2} \) |
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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
−10.46110039864845255293462895732, −9.730267574420325740589210375175, −9.218924549307231840933372186383, −7.42288592828611569139328357492, −6.79071395632317220731356604531, −5.12575520538648091855957770060, −3.84411263116080442237939057130, −3.51378286914115127078644807307, −1.72919836255097088158160211516, −0.70166395548283325072369679538,
1.37856277933832828852985977666, 2.40642332623097353548037067330, 4.20250611857900345369555922285, 5.58792442969632903347865325850, 6.54695905359745573786327372048, 6.86827011961163260066104223822, 8.251932034365359853323552101508, 8.685109261953246114143489033067, 9.976513830803770978630269207724, 11.35823990547219859818358347985
