| L(s) = 1 | + (−1.15 + 0.811i)2-s + (1.70 − 0.300i)3-s + (0.684 − 1.87i)4-s + (0.651 − 7.44i)5-s + (−1.73 + 1.73i)6-s + (2.81 − 2.35i)7-s + (0.732 + 2.73i)8-s + (2.81 − 1.02i)9-s + (5.28 + 9.14i)10-s + (−13.9 − 8.04i)11-s + (0.601 − 3.41i)12-s + (−19.7 + 9.20i)13-s + (−1.34 + 5.01i)14-s + (−1.12 − 12.8i)15-s + (−3.06 − 2.57i)16-s + (5.50 − 11.8i)17-s + ⋯ |
| L(s) = 1 | + (−0.579 + 0.405i)2-s + (0.568 − 0.100i)3-s + (0.171 − 0.469i)4-s + (0.130 − 1.48i)5-s + (−0.288 + 0.288i)6-s + (0.401 − 0.337i)7-s + (0.0915 + 0.341i)8-s + (0.313 − 0.114i)9-s + (0.528 + 0.914i)10-s + (−1.26 − 0.731i)11-s + (0.0501 − 0.284i)12-s + (−1.51 + 0.708i)13-s + (−0.0959 + 0.358i)14-s + (−0.0751 − 0.859i)15-s + (−0.191 − 0.160i)16-s + (0.324 − 0.695i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 222 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0668 + 0.997i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 222 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0668 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.784265 - 0.838573i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.784265 - 0.838573i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.15 - 0.811i)T \) |
| 3 | \( 1 + (-1.70 + 0.300i)T \) |
| 37 | \( 1 + (-4.81 + 36.6i)T \) |
| good | 5 | \( 1 + (-0.651 + 7.44i)T + (-24.6 - 4.34i)T^{2} \) |
| 7 | \( 1 + (-2.81 + 2.35i)T + (8.50 - 48.2i)T^{2} \) |
| 11 | \( 1 + (13.9 + 8.04i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (19.7 - 9.20i)T + (108. - 129. i)T^{2} \) |
| 17 | \( 1 + (-5.50 + 11.8i)T + (-185. - 221. i)T^{2} \) |
| 19 | \( 1 + (8.56 + 5.99i)T + (123. + 339. i)T^{2} \) |
| 23 | \( 1 + (-3.79 - 1.01i)T + (458. + 264.5i)T^{2} \) |
| 29 | \( 1 + (-29.9 + 8.01i)T + (728. - 420.5i)T^{2} \) |
| 31 | \( 1 + (-42.3 - 42.3i)T + 961iT^{2} \) |
| 41 | \( 1 + (-19.3 + 53.0i)T + (-1.28e3 - 1.08e3i)T^{2} \) |
| 43 | \( 1 + (-25.7 + 25.7i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (13.6 + 23.6i)T + (-1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (43.8 + 36.8i)T + (487. + 2.76e3i)T^{2} \) |
| 59 | \( 1 + (-45.9 + 4.01i)T + (3.42e3 - 604. i)T^{2} \) |
| 61 | \( 1 + (2.23 + 4.78i)T + (-2.39e3 + 2.85e3i)T^{2} \) |
| 67 | \( 1 + (-73.8 - 88.0i)T + (-779. + 4.42e3i)T^{2} \) |
| 71 | \( 1 + (-4.09 - 23.2i)T + (-4.73e3 + 1.72e3i)T^{2} \) |
| 73 | \( 1 - 71.5iT - 5.32e3T^{2} \) |
| 79 | \( 1 + (-12.1 + 138. i)T + (-6.14e3 - 1.08e3i)T^{2} \) |
| 83 | \( 1 + (-36.8 + 13.4i)T + (5.27e3 - 4.42e3i)T^{2} \) |
| 89 | \( 1 + (-9.62 - 109. i)T + (-7.80e3 + 1.37e3i)T^{2} \) |
| 97 | \( 1 + (-55.8 - 14.9i)T + (8.14e3 + 4.70e3i)T^{2} \) |
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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
−11.92326107865393722264605572714, −10.53334669872216799157530469324, −9.580536410216564480044424200096, −8.700942298191854128524674631938, −8.018336585273059557436883912865, −7.04025660596294445893605930842, −5.32740198580273708258290254548, −4.61867263458181229948984443620, −2.41989144798744057771634015732, −0.68966165402716249120447760054,
2.35045551735414628530080969527, 2.93066303749321461576644768224, 4.74664026884852739895644042800, 6.42157784495486245654634162320, 7.69829814178878534301907840799, 8.053001666334956138509254224801, 9.842169559456574252771666075890, 10.12689038094803550923068747413, 11.03887199245955992986512671476, 12.22893377577175938459908257720
