L(s) = 1 | + (0.190 − 0.109i)2-s + (−0.800 − 1.38i)3-s + (−0.975 + 1.69i)4-s + (−0.304 − 0.175i)6-s + (0.287 + 0.166i)7-s + 0.868i·8-s + (0.219 − 0.380i)9-s + (4.65 − 2.68i)11-s + 3.12·12-s + (3.55 + 0.619i)13-s + 0.0729·14-s + (−1.85 − 3.21i)16-s + (2.53 − 4.38i)17-s − 0.0965i·18-s + (−1.96 − 1.13i)19-s + ⋯ |
L(s) = 1 | + (0.134 − 0.0776i)2-s + (−0.461 − 0.800i)3-s + (−0.487 + 0.845i)4-s + (−0.124 − 0.0717i)6-s + (0.108 + 0.0627i)7-s + 0.306i·8-s + (0.0732 − 0.126i)9-s + (1.40 − 0.809i)11-s + 0.901·12-s + (0.985 + 0.171i)13-s + 0.0195·14-s + (−0.464 − 0.803i)16-s + (0.614 − 1.06i)17-s − 0.0227i·18-s + (−0.450 − 0.260i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.758 + 0.651i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.758 + 0.651i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.13579 - 0.420459i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.13579 - 0.420459i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 13 | \( 1 + (-3.55 - 0.619i)T \) |
good | 2 | \( 1 + (-0.190 + 0.109i)T + (1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (0.800 + 1.38i)T + (-1.5 + 2.59i)T^{2} \) |
| 7 | \( 1 + (-0.287 - 0.166i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-4.65 + 2.68i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-2.53 + 4.38i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.96 + 1.13i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.41 - 2.45i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.45 - 2.51i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 5.46iT - 31T^{2} \) |
| 37 | \( 1 + (-5.17 + 2.98i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.23 + 1.86i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.53 + 4.38i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 8.34iT - 47T^{2} \) |
| 53 | \( 1 - 1.56T + 53T^{2} \) |
| 59 | \( 1 + (-2.34 - 1.35i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (7.05 - 12.2i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (8.94 - 5.16i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (11.0 + 6.39i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 9.68iT - 73T^{2} \) |
| 79 | \( 1 - 4.51T + 79T^{2} \) |
| 83 | \( 1 - 4.26iT - 83T^{2} \) |
| 89 | \( 1 + (2.79 - 1.61i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (2.17 + 1.25i)T + (48.5 + 84.0i)T^{2} \) |
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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
−11.79083025578178371713992585860, −10.97014948603295767038646044604, −9.283621826418644951479118433769, −8.781747788272193104768050760309, −7.56496897001543181646387689833, −6.71218470163702905693252296840, −5.71536898338905352541957669621, −4.23080252845427046761841712341, −3.20155789960549266813163881591, −1.12229938892649134080325827589,
1.48976487900079582184095432555, 3.98810809267909877390251759501, 4.50216255280512620793232077109, 5.79970666658670666604016268561, 6.46978033144148622607673017791, 8.056789070694168657191342251545, 9.213227182567196710551241975687, 9.930184925009856571284532086787, 10.66761696435859452710950017075, 11.45919896105698690873856739631