| L(s) = 1 | + 8.43i·3-s − 14.7·5-s + 70.0·7-s + 9.83·9-s + 37.6i·11-s + 157. i·13-s − 124. i·15-s + 461. i·17-s − 690.·19-s + 590. i·21-s − 345. i·23-s − 408.·25-s + 766. i·27-s − 1.45e3i·29-s + (420. + 864. i)31-s + ⋯ |
| L(s) = 1 | + 0.937i·3-s − 0.588·5-s + 1.42·7-s + 0.121·9-s + 0.311i·11-s + 0.929i·13-s − 0.551i·15-s + 1.59i·17-s − 1.91·19-s + 1.33i·21-s − 0.652i·23-s − 0.653·25-s + 1.05i·27-s − 1.73i·29-s + (0.437 + 0.899i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.437 - 0.899i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.437 - 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.851981 + 1.36247i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.851981 + 1.36247i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 31 | \( 1 + (-420. - 864. i)T \) |
| good | 3 | \( 1 - 8.43iT - 81T^{2} \) |
| 5 | \( 1 + 14.7T + 625T^{2} \) |
| 7 | \( 1 - 70.0T + 2.40e3T^{2} \) |
| 11 | \( 1 - 37.6iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 157. iT - 2.85e4T^{2} \) |
| 17 | \( 1 - 461. iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 690.T + 1.30e5T^{2} \) |
| 23 | \( 1 + 345. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 1.45e3iT - 7.07e5T^{2} \) |
| 37 | \( 1 - 2.39e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 - 2.38e3T + 2.82e6T^{2} \) |
| 43 | \( 1 - 1.99e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 - 1.37e3T + 4.87e6T^{2} \) |
| 53 | \( 1 - 413. iT - 7.89e6T^{2} \) |
| 59 | \( 1 - 4.58e3T + 1.21e7T^{2} \) |
| 61 | \( 1 + 5.97e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 2.78e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + 4.20e3T + 2.54e7T^{2} \) |
| 73 | \( 1 - 2.33e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + 2.77e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + 402. iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 1.09e4iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 9.44e3T + 8.85e7T^{2} \) |
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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
−12.94469737094787917575913614244, −11.77913133870094264042538817537, −10.90970033337328050580888645007, −10.09125405617353268237924129043, −8.666803440068747521753674698696, −7.922724647602121096711038488808, −6.32894681642379923548247955339, −4.46961884171268848193165269879, −4.23598389579232752053403421203, −1.85116505600354256105250398079,
0.70508037484400056570609170393, 2.23152759880245786631432386610, 4.22363333909705277466061567965, 5.58199187240048074297434303260, 7.21060165170534424665562560432, 7.81725545713514670034798995822, 8.847149576301021304285582895486, 10.59011643700216585274054891964, 11.45665529140766260472584237533, 12.35506384445067192401533407053
