| 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
−11.35823990547219859818358347985, −9.976513830803770978630269207724, −8.685109261953246114143489033067, −8.251932034365359853323552101508, −6.86827011961163260066104223822, −6.54695905359745573786327372048, −5.58792442969632903347865325850, −4.20250611857900345369555922285, −2.40642332623097353548037067330, −1.37856277933832828852985977666,
0.70166395548283325072369679538, 1.72919836255097088158160211516, 3.51378286914115127078644807307, 3.84411263116080442237939057130, 5.12575520538648091855957770060, 6.79071395632317220731356604531, 7.42288592828611569139328357492, 9.218924549307231840933372186383, 9.730267574420325740589210375175, 10.46110039864845255293462895732
