| L(s) = 1 | + (2.26 + 5.18i)2-s + 10.8·3-s + (−21.7 + 23.5i)4-s + (24.5 + 56.0i)6-s + 163. i·7-s + (−171. − 59.0i)8-s − 125.·9-s + 321. i·11-s + (−234. + 254. i)12-s + 128.·13-s + (−848. + 371. i)14-s + (−82.2 − 1.02e3i)16-s − 2.11e3i·17-s + (−285. − 652. i)18-s + 1.45e3i·19-s + ⋯ |
| L(s) = 1 | + (0.401 + 0.916i)2-s + 0.694·3-s + (−0.678 + 0.734i)4-s + (0.278 + 0.636i)6-s + 1.26i·7-s + (−0.945 − 0.326i)8-s − 0.517·9-s + 0.801i·11-s + (−0.470 + 0.510i)12-s + 0.210·13-s + (−1.15 + 0.506i)14-s + (−0.0802 − 0.996i)16-s − 1.77i·17-s + (−0.207 − 0.474i)18-s + 0.924i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.714 + 0.699i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.714 + 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.300964614\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.300964614\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-2.26 - 5.18i)T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 10.8T + 243T^{2} \) |
| 7 | \( 1 - 163. iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 321. iT - 1.61e5T^{2} \) |
| 13 | \( 1 - 128.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 2.11e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 1.45e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 1.23e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 4.07e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 3.95e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.06e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 5.90e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.64e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.32e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 3.06e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.52e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 + 3.91e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 + 2.08e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.38e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.34e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 1.25e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 6.68e3T + 3.93e9T^{2} \) |
| 89 | \( 1 - 9.04e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.49e5iT - 8.58e9T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.23496557473850210454877950171, −11.66510531502031684396230529673, −9.697324168280142761642526189905, −9.016199664545290395844844671756, −8.150317458606447658435790845403, −7.16523698427894301825664912163, −5.85282021063144197989246061862, −5.00876087010191953244825179749, −3.45886148824141637088398856703, −2.34245738577102403587805900454,
0.29892840858045812443097077379, 1.73177391494001697548252318619, 3.25985570314821737261539487327, 3.93696621712279935068480356125, 5.40563477699705896066184334450, 6.74359292808025765901059854925, 8.326126867819585433700272083102, 8.919490553403919379131579907612, 10.37405134321371750827491332120, 10.79142544147016221898123491018
