L(s) = 1 | + (−0.0939 + 5.19i)3-s + (−7.82 − 13.5i)5-s + (17.6 + 5.69i)7-s + (−26.9 − 0.976i)9-s + (34.2 + 19.7i)11-s + (−55.5 + 32.0i)13-s + (71.1 − 39.3i)15-s − 56.6·17-s + 117. i·19-s + (−31.2 + 91.0i)21-s + (−6.59 + 3.80i)23-s + (−59.9 + 103. i)25-s + (7.61 − 140. i)27-s + (39.8 + 22.9i)29-s + (−251. + 145. i)31-s + ⋯ |
L(s) = 1 | + (−0.0180 + 0.999i)3-s + (−0.699 − 1.21i)5-s + (0.951 + 0.307i)7-s + (−0.999 − 0.0361i)9-s + (0.937 + 0.541i)11-s + (−1.18 + 0.684i)13-s + (1.22 − 0.677i)15-s − 0.808·17-s + 1.41i·19-s + (−0.324 + 0.945i)21-s + (−0.0597 + 0.0345i)23-s + (−0.479 + 0.830i)25-s + (0.0542 − 0.998i)27-s + (0.254 + 0.147i)29-s + (−1.45 + 0.842i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 - 0.499i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.866 - 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.8388473148\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8388473148\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.0939 - 5.19i)T \) |
| 7 | \( 1 + (-17.6 - 5.69i)T \) |
good | 5 | \( 1 + (7.82 + 13.5i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (-34.2 - 19.7i)T + (665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (55.5 - 32.0i)T + (1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 + 56.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 117. iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (6.59 - 3.80i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-39.8 - 22.9i)T + (1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + (251. - 145. i)T + (1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + 335.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (97.2 + 168. i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-152. + 263. i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (318. - 550. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + 274. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (-258. - 448. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (142. + 82.1i)T + (1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-368. - 637. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 599. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 214. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + (454. - 786. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-389. + 675. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 - 443.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (337. + 194. i)T + (4.56e5 + 7.90e5i)T^{2} \) |
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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
−12.03087263705918090707198599734, −11.18209125928974428377620533373, −9.961129178187092770525234827003, −8.964303728580934482409587802312, −8.478746539519050299269480533235, −7.19306469099824884144841875389, −5.44238305611079895638755308890, −4.62736077599956382540843051837, −3.91724396936898969156372613849, −1.78134471406512378655301413150,
0.32315610590414050690994258894, 2.16157171640329378441201629974, 3.41567308810965334955310305407, 4.98720061026271581850407241986, 6.52258723181846063646057404025, 7.20909373233592991013254373563, 7.946274032730471111074802796817, 9.058578406915813341996553943043, 10.66365708871162214410310642301, 11.33877529153766235334256774129