| L(s) = 1 | + (−1.23 − 3.80i)2-s + (−12.9 + 9.40i)4-s + (71.9 + 23.3i)5-s + (120. + 165. i)7-s + (51.7 + 37.6i)8-s − 302. i·10-s + (330. − 227. i)11-s + (83.5 − 27.1i)13-s + (481. − 663. i)14-s + (79.1 − 243. i)16-s + (−246. + 759. i)17-s + (−1.63e3 + 2.24e3i)19-s + (−1.15e3 + 374. i)20-s + (−1.27e3 − 975. i)22-s + 2.43e3i·23-s + ⋯ |
| L(s) = 1 | + (−0.218 − 0.672i)2-s + (−0.404 + 0.293i)4-s + (1.28 + 0.418i)5-s + (0.929 + 1.27i)7-s + (0.286 + 0.207i)8-s − 0.957i·10-s + (0.823 − 0.567i)11-s + (0.137 − 0.0445i)13-s + (0.657 − 0.904i)14-s + (0.0772 − 0.237i)16-s + (−0.207 + 0.637i)17-s + (−1.03 + 1.42i)19-s + (−0.643 + 0.209i)20-s + (−0.561 − 0.429i)22-s + 0.960i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.802 - 0.596i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.802 - 0.596i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.333546190\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.333546190\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.23 + 3.80i)T \) |
| 3 | \( 1 \) |
| 11 | \( 1 + (-330. + 227. i)T \) |
| good | 5 | \( 1 + (-71.9 - 23.3i)T + (2.52e3 + 1.83e3i)T^{2} \) |
| 7 | \( 1 + (-120. - 165. i)T + (-5.19e3 + 1.59e4i)T^{2} \) |
| 13 | \( 1 + (-83.5 + 27.1i)T + (3.00e5 - 2.18e5i)T^{2} \) |
| 17 | \( 1 + (246. - 759. i)T + (-1.14e6 - 8.34e5i)T^{2} \) |
| 19 | \( 1 + (1.63e3 - 2.24e3i)T + (-7.65e5 - 2.35e6i)T^{2} \) |
| 23 | \( 1 - 2.43e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + (3.76e3 - 2.73e3i)T + (6.33e6 - 1.95e7i)T^{2} \) |
| 31 | \( 1 + (3.02e3 + 9.30e3i)T + (-2.31e7 + 1.68e7i)T^{2} \) |
| 37 | \( 1 + (-4.14e3 + 3.01e3i)T + (2.14e7 - 6.59e7i)T^{2} \) |
| 41 | \( 1 + (-6.21e3 - 4.51e3i)T + (3.58e7 + 1.10e8i)T^{2} \) |
| 43 | \( 1 + 5.54e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + (6.76e3 - 9.30e3i)T + (-7.08e7 - 2.18e8i)T^{2} \) |
| 53 | \( 1 + (-1.11e4 + 3.63e3i)T + (3.38e8 - 2.45e8i)T^{2} \) |
| 59 | \( 1 + (1.79e3 + 2.46e3i)T + (-2.20e8 + 6.79e8i)T^{2} \) |
| 61 | \( 1 + (-3.86e4 - 1.25e4i)T + (6.83e8 + 4.96e8i)T^{2} \) |
| 67 | \( 1 + 4.47e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (-6.21e4 - 2.01e4i)T + (1.45e9 + 1.06e9i)T^{2} \) |
| 73 | \( 1 + (3.24e4 + 4.47e4i)T + (-6.40e8 + 1.97e9i)T^{2} \) |
| 79 | \( 1 + (-9.65e4 + 3.13e4i)T + (2.48e9 - 1.80e9i)T^{2} \) |
| 83 | \( 1 + (2.70e4 - 8.31e4i)T + (-3.18e9 - 2.31e9i)T^{2} \) |
| 89 | \( 1 + 1.47e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 + (-3.01e4 - 9.26e4i)T + (-6.94e9 + 5.04e9i)T^{2} \) |
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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
−11.52955016658333660369448960718, −10.82519777443876536540301276238, −9.670883275030412001674844747572, −8.946461792190250899931333324359, −7.978253474210307646629978518604, −6.13479004218205549136220818316, −5.57341143599442086970350217905, −3.85690802890368060916566245503, −2.24527840734686973146382079697, −1.60693626461122897300740966952,
0.793416437691105861134488589596, 1.96326220831555676393762554873, 4.30415499989732923149034451068, 5.07617426598625164454284114697, 6.49737193393142403122645373814, 7.21365616817546010761095875987, 8.578805234713107021107101073721, 9.376878308795525125115730115981, 10.36192080262539579430515251068, 11.25220599381932390095193616610
