L(s) = 1 | + (1.13 − 2.59i)2-s + (5.07 + 1.13i)3-s + (−5.43 − 5.86i)4-s + (−11.2 − 11.2i)5-s + (8.68 − 11.8i)6-s + 30.2·7-s + (−21.3 + 7.45i)8-s + (24.4 + 11.5i)9-s + (−41.9 + 16.4i)10-s + (−14.9 + 14.9i)11-s + (−20.9 − 35.9i)12-s + (10.4 + 10.4i)13-s + (34.1 − 78.3i)14-s + (−44.3 − 69.9i)15-s + (−4.87 + 63.8i)16-s + 69.5i·17-s + ⋯ |
L(s) = 1 | + (0.400 − 0.916i)2-s + (0.975 + 0.218i)3-s + (−0.679 − 0.733i)4-s + (−1.00 − 1.00i)5-s + (0.590 − 0.806i)6-s + 1.63·7-s + (−0.944 + 0.329i)8-s + (0.904 + 0.426i)9-s + (−1.32 + 0.520i)10-s + (−0.410 + 0.410i)11-s + (−0.502 − 0.864i)12-s + (0.223 + 0.223i)13-s + (0.652 − 1.49i)14-s + (−0.763 − 1.20i)15-s + (−0.0761 + 0.997i)16-s + 0.992i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0960 + 0.995i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0960 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.42111 - 1.29053i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.42111 - 1.29053i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.13 + 2.59i)T \) |
| 3 | \( 1 + (-5.07 - 1.13i)T \) |
good | 5 | \( 1 + (11.2 + 11.2i)T + 125iT^{2} \) |
| 7 | \( 1 - 30.2T + 343T^{2} \) |
| 11 | \( 1 + (14.9 - 14.9i)T - 1.33e3iT^{2} \) |
| 13 | \( 1 + (-10.4 - 10.4i)T + 2.19e3iT^{2} \) |
| 17 | \( 1 - 69.5iT - 4.91e3T^{2} \) |
| 19 | \( 1 + (36.4 - 36.4i)T - 6.85e3iT^{2} \) |
| 23 | \( 1 - 1.28iT - 1.21e4T^{2} \) |
| 29 | \( 1 + (-119. + 119. i)T - 2.43e4iT^{2} \) |
| 31 | \( 1 + 172. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + (235. - 235. i)T - 5.06e4iT^{2} \) |
| 41 | \( 1 + 19.9T + 6.89e4T^{2} \) |
| 43 | \( 1 + (294. + 294. i)T + 7.95e4iT^{2} \) |
| 47 | \( 1 + 69.5T + 1.03e5T^{2} \) |
| 53 | \( 1 + (122. + 122. i)T + 1.48e5iT^{2} \) |
| 59 | \( 1 + (-63.2 + 63.2i)T - 2.05e5iT^{2} \) |
| 61 | \( 1 + (352. + 352. i)T + 2.26e5iT^{2} \) |
| 67 | \( 1 + (228. - 228. i)T - 3.00e5iT^{2} \) |
| 71 | \( 1 - 524. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 578. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 745. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + (-286. - 286. i)T + 5.71e5iT^{2} \) |
| 89 | \( 1 + 203.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 39.4T + 9.12e5T^{2} \) |
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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
−14.83799940616768790375892509940, −13.68425139640393975082480997252, −12.52989284040409001587525263937, −11.53285746159668945799892871617, −10.24473387428427398804165280525, −8.587219399339664807108434498721, −8.079229289481370271792489482318, −4.87568048667022032470413173932, −4.02855920733033556038425290171, −1.72809329767637566374055517354,
3.16591804439138827737960065382, 4.74116660450628918308283819244, 6.99834432676598443049783402594, 7.85479146488833698649710659351, 8.700177779290641662756222528759, 10.85376063851485046466285783783, 12.11178373917899414960010154588, 13.70572525739049597907736287677, 14.46772166849411720954635320665, 15.16450493180902954097661752509