L(s) = 1 | + (−3.66 − 3.68i)3-s + (8.29 − 14.3i)5-s + (18.2 + 3.26i)7-s + (−0.127 + 26.9i)9-s + (46.2 − 26.7i)11-s + (11.0 + 6.37i)13-s + (−83.3 + 22.1i)15-s + 96.7·17-s + 54.6i·19-s + (−54.7 − 79.1i)21-s + (−55.6 − 32.1i)23-s + (−75.2 − 130. i)25-s + (99.9 − 98.5i)27-s + (−112. + 65.0i)29-s + (−190. − 110. i)31-s + ⋯ |
L(s) = 1 | + (−0.705 − 0.708i)3-s + (0.742 − 1.28i)5-s + (0.984 + 0.176i)7-s + (−0.00470 + 0.999i)9-s + (1.26 − 0.732i)11-s + (0.235 + 0.136i)13-s + (−1.43 + 0.380i)15-s + 1.38·17-s + 0.660i·19-s + (−0.569 − 0.822i)21-s + (−0.504 − 0.291i)23-s + (−0.601 − 1.04i)25-s + (0.712 − 0.702i)27-s + (−0.721 + 0.416i)29-s + (−1.10 − 0.637i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00752 + 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.00752 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.948444006\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.948444006\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (3.66 + 3.68i)T \) |
| 7 | \( 1 + (-18.2 - 3.26i)T \) |
good | 5 | \( 1 + (-8.29 + 14.3i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (-46.2 + 26.7i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-11.0 - 6.37i)T + (1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 - 96.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 54.6iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (55.6 + 32.1i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (112. - 65.0i)T + (1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (190. + 110. i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 - 279.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-185. + 320. i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (153. + 266. i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (163. + 283. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 - 451. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (258. - 448. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (234. - 135. i)T + (1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (370. - 642. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 914. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 337. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + (498. + 863. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (17.0 + 29.4i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 - 208.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (1.10e3 - 635. i)T + (4.56e5 - 7.90e5i)T^{2} \) |
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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
−11.68449624366177282848611560085, −10.55059821184077238919626852764, −9.256865494646707848048175035269, −8.454395774222029523134995691531, −7.45161242152376014618505373796, −5.85128949052282692955924158408, −5.56010476760519687975192653848, −4.16834211901894769869899292946, −1.75801891914165725095384220138, −1.00494023476183953793632332979,
1.53940847832826708974301499879, 3.33305794496782911854717504164, 4.57697581966275481408473480020, 5.80348902803805156188960567271, 6.64204653982290673300766831741, 7.73260387792179724719270982890, 9.430172855165562636537912447198, 9.911528632979879436766550627514, 11.05775759636641677157878525458, 11.40390465159288643697480616515