L(s) = 1 | + (−4.67 + 2.70i)2-s + (2.75 − 4.40i)3-s + (10.5 − 18.3i)4-s + (−3.43 + 5.95i)5-s + (−1.00 + 28.0i)6-s + (−17.3 + 6.49i)7-s + 71.1i·8-s + (−11.8 − 24.2i)9-s − 37.1i·10-s + (−10.9 + 6.30i)11-s + (−51.5 − 97.2i)12-s + (−22.5 − 13.0i)13-s + (63.5 − 77.2i)14-s + (16.7 + 31.5i)15-s + (−107. − 186. i)16-s − 124.·17-s + ⋯ |
L(s) = 1 | + (−1.65 + 0.954i)2-s + (0.530 − 0.847i)3-s + (1.32 − 2.29i)4-s + (−0.307 + 0.532i)5-s + (−0.0680 + 1.90i)6-s + (−0.936 + 0.350i)7-s + 3.14i·8-s + (−0.437 − 0.899i)9-s − 1.17i·10-s + (−0.299 + 0.172i)11-s + (−1.24 − 2.33i)12-s + (−0.481 − 0.277i)13-s + (1.21 − 1.47i)14-s + (0.288 + 0.543i)15-s + (−1.68 − 2.91i)16-s − 1.77·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 + 0.445i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.895 + 0.445i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.00464086 - 0.0197543i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.00464086 - 0.0197543i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-2.75 + 4.40i)T \) |
| 7 | \( 1 + (17.3 - 6.49i)T \) |
good | 2 | \( 1 + (4.67 - 2.70i)T + (4 - 6.92i)T^{2} \) |
| 5 | \( 1 + (3.43 - 5.95i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (10.9 - 6.30i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (22.5 + 13.0i)T + (1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + 124.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 41.3iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (97.6 + 56.4i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-114. + 66.3i)T + (1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (155. + 89.5i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 - 148.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (93.6 - 162. i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-99.1 - 171. i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-92.1 - 159. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + 359. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (-182. + 315. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-300. + 173. i)T + (1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (182. - 315. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + 565. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 737. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + (451. + 782. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (382. + 662. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + 395.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-243. + 140. i)T + (4.56e5 - 7.90e5i)T^{2} \) |
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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
−14.50605065422730675544151800872, −12.97889266164736971660934437320, −11.40512655668791905281766102354, −10.05105395770353641284713003339, −9.037798753516661471048038449914, −7.981953188353379976754907384293, −6.95454410151386757436675259464, −6.14693359573326964028631698988, −2.38239216458332558963287310910, −0.02024898086410944749359484021,
2.55874626847377763625797773316, 4.08069182292693136523859557114, 7.11407805940947280732072785891, 8.520262752921377355643448237432, 9.194246107683991261715086122550, 10.19744575265368768131390910579, 11.06391062824174314816201724183, 12.35251348688042807489012803883, 13.52035460222902345109813186214, 15.62225991199584265930578859494