| L(s) = 1 | − 10.7·2-s + (9.69 − 12.2i)3-s + 83.5·4-s − 33.1i·5-s + (−104. + 131. i)6-s − 174.·7-s − 554.·8-s + (−55.0 − 236. i)9-s + 356. i·10-s − 67.7·11-s + (809. − 1.01e3i)12-s + 762. i·13-s + 1.87e3·14-s + (−404. − 321. i)15-s + 3.28e3·16-s + 1.05e3i·17-s + ⋯ |
| L(s) = 1 | − 1.90·2-s + (0.621 − 0.783i)3-s + 2.61·4-s − 0.592i·5-s + (−1.18 + 1.48i)6-s − 1.34·7-s − 3.06·8-s + (−0.226 − 0.973i)9-s + 1.12i·10-s − 0.168·11-s + (1.62 − 2.04i)12-s + 1.25i·13-s + 2.55·14-s + (−0.464 − 0.368i)15-s + 3.20·16-s + 0.885i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0152i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0152i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.6354810683\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6354810683\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-9.69 + 12.2i)T \) |
| 59 | \( 1 + (1.63e4 + 2.11e4i)T \) |
| good | 2 | \( 1 + 10.7T + 32T^{2} \) |
| 5 | \( 1 + 33.1iT - 3.12e3T^{2} \) |
| 7 | \( 1 + 174.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 67.7T + 1.61e5T^{2} \) |
| 13 | \( 1 - 762. iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 1.05e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 1.51e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.50e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.78e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 4.00e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 - 6.21e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 1.46e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 + 1.99e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 1.74e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.26e3iT - 4.18e8T^{2} \) |
| 61 | \( 1 - 2.18e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 - 2.03e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 2.64e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + 7.84e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 7.81e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.68e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 9.32e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 4.25e4iT - 8.58e9T^{2} \) |
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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
−11.77641918625657604045521196593, −10.37607104295194315646889031015, −9.359625465771719449262105252748, −8.956033477210483880674678358884, −7.898729740519950914711876828594, −6.91294826581116896441359052142, −6.18492399927767484060215308638, −3.32695341259541031564148959637, −1.99256510150803695141689787469, −0.839463523494928636465733707483,
0.47457046960579524913804026083, 2.60293329594222397418770620572, 3.24554209709724935132998476930, 5.78556004774771078638165486979, 7.08974255251241696682639787983, 7.88610818897004346819899561571, 9.030773105698321449293994785246, 9.781260817865596246226429694910, 10.31387135411773433415616383592, 11.16172775376604786806657907314
