| L(s) = 1 | + (−1.05 + 0.946i)2-s + (0.342 − 0.939i)3-s + (0.209 − 1.98i)4-s + (2.68 − 0.473i)5-s + (0.529 + 1.31i)6-s + (0.416 − 0.720i)7-s + (1.66 + 2.28i)8-s + (−0.766 − 0.642i)9-s + (−2.37 + 3.04i)10-s + (1.90 − 1.09i)11-s + (−1.79 − 0.877i)12-s + (0.794 + 2.18i)13-s + (0.244 + 1.15i)14-s + (0.473 − 2.68i)15-s + (−3.91 − 0.833i)16-s + (1.80 − 1.51i)17-s + ⋯ |
| L(s) = 1 | + (−0.743 + 0.669i)2-s + (0.197 − 0.542i)3-s + (0.104 − 0.994i)4-s + (1.20 − 0.211i)5-s + (0.216 + 0.535i)6-s + (0.157 − 0.272i)7-s + (0.587 + 0.809i)8-s + (−0.255 − 0.214i)9-s + (−0.751 + 0.961i)10-s + (0.573 − 0.331i)11-s + (−0.518 − 0.253i)12-s + (0.220 + 0.605i)13-s + (0.0653 + 0.307i)14-s + (0.122 − 0.693i)15-s + (−0.978 − 0.208i)16-s + (0.438 − 0.368i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 + 0.324i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.945 + 0.324i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.29175 - 0.215440i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.29175 - 0.215440i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.05 - 0.946i)T \) |
| 3 | \( 1 + (-0.342 + 0.939i)T \) |
| 19 | \( 1 + (3.91 - 1.90i)T \) |
| good | 5 | \( 1 + (-2.68 + 0.473i)T + (4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (-0.416 + 0.720i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.90 + 1.09i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.794 - 2.18i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-1.80 + 1.51i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (-1.13 + 6.45i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (1.10 - 1.31i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-2.45 + 4.25i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 1.34iT - 37T^{2} \) |
| 41 | \( 1 + (-7.26 - 2.64i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-1.29 + 0.229i)T + (40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (4.64 + 3.89i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (-2.52 - 0.444i)T + (49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (-3.70 - 4.42i)T + (-10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-1.61 - 0.284i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (10.2 - 12.2i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (1.67 + 9.50i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (7.83 + 2.85i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (-0.702 - 0.255i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-1.96 - 1.13i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-0.0971 + 0.0353i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (5.55 - 4.66i)T + (16.8 - 95.5i)T^{2} \) |
| 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
−10.75565649435021676617467701621, −9.912234137205087618352848007301, −9.072593240710798388113020017956, −8.433923826994077605628023242559, −7.31834169729890631465734772415, −6.35748563127705807726926629362, −5.83321561711037224253831517397, −4.45004567754415957998955146585, −2.34151204553162490728567487729, −1.18562029952486578177250016289,
1.64560394620749512352967403122, 2.78759265986556294043169244160, 4.00159921501683441871356938778, 5.40622417155463276970377261662, 6.50294076063722975438991422102, 7.72930579146998473655071558866, 8.786915888225925389045214758522, 9.436440366673149494249686771935, 10.15674723062927855877244859418, 10.82973641654608608700176781966
