| L(s) = 1 | + (−0.415 + 0.909i)3-s + (−3.13 − 0.920i)5-s + (−2.56 + 0.493i)7-s + (−0.654 − 0.755i)9-s + (−1.04 + 0.999i)11-s + (−0.0752 + 1.57i)13-s + (2.13 − 2.46i)15-s + (6.09 − 4.79i)17-s + (6.41 + 1.23i)19-s + (0.614 − 2.53i)21-s + (2.94 − 0.281i)23-s + (4.76 + 3.06i)25-s + (0.959 − 0.281i)27-s + (2.68 − 4.65i)29-s + (−0.0686 − 1.44i)31-s + ⋯ |
| L(s) = 1 | + (−0.239 + 0.525i)3-s + (−1.40 − 0.411i)5-s + (−0.968 + 0.186i)7-s + (−0.218 − 0.251i)9-s + (−0.315 + 0.301i)11-s + (−0.0208 + 0.438i)13-s + (0.552 − 0.637i)15-s + (1.47 − 1.16i)17-s + (1.47 + 0.283i)19-s + (0.134 − 0.553i)21-s + (0.614 − 0.0586i)23-s + (0.953 + 0.612i)25-s + (0.184 − 0.0542i)27-s + (0.499 − 0.865i)29-s + (−0.0123 − 0.258i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 + 0.326i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.945 + 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.837575 - 0.140482i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.837575 - 0.140482i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.415 - 0.909i)T \) |
| 67 | \( 1 + (1.16 - 8.10i)T \) |
| good | 5 | \( 1 + (3.13 + 0.920i)T + (4.20 + 2.70i)T^{2} \) |
| 7 | \( 1 + (2.56 - 0.493i)T + (6.49 - 2.60i)T^{2} \) |
| 11 | \( 1 + (1.04 - 0.999i)T + (0.523 - 10.9i)T^{2} \) |
| 13 | \( 1 + (0.0752 - 1.57i)T + (-12.9 - 1.23i)T^{2} \) |
| 17 | \( 1 + (-6.09 + 4.79i)T + (4.00 - 16.5i)T^{2} \) |
| 19 | \( 1 + (-6.41 - 1.23i)T + (17.6 + 7.06i)T^{2} \) |
| 23 | \( 1 + (-2.94 + 0.281i)T + (22.5 - 4.35i)T^{2} \) |
| 29 | \( 1 + (-2.68 + 4.65i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (0.0686 + 1.44i)T + (-30.8 + 2.94i)T^{2} \) |
| 37 | \( 1 + (0.917 + 1.58i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (0.463 + 0.185i)T + (29.6 + 28.2i)T^{2} \) |
| 43 | \( 1 + (-0.294 - 2.04i)T + (-41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 + (4.84 + 6.80i)T + (-15.3 + 44.4i)T^{2} \) |
| 53 | \( 1 + (-1.33 + 9.30i)T + (-50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (0.211 - 0.135i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (-5.09 - 4.86i)T + (2.90 + 60.9i)T^{2} \) |
| 71 | \( 1 + (-0.989 - 0.777i)T + (16.7 + 68.9i)T^{2} \) |
| 73 | \( 1 + (-1.98 - 1.89i)T + (3.47 + 72.9i)T^{2} \) |
| 79 | \( 1 + (-12.0 + 6.20i)T + (45.8 - 64.3i)T^{2} \) |
| 83 | \( 1 + (-3.33 - 13.7i)T + (-73.7 + 38.0i)T^{2} \) |
| 89 | \( 1 + (3.98 + 8.71i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (6.62 + 11.4i)T + (-48.5 + 84.0i)T^{2} \) |
| 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
−9.883131115006132003665552737243, −9.664471452970627168869166936593, −8.505572946969820381244392693679, −7.62352741659350924205274204771, −6.92249192919842181569967074809, −5.58551601488562817669409970968, −4.81092383948070302499081008262, −3.71042151153931174202246867926, −3.01022997764526683564798868442, −0.61994167533869470463181540667,
0.935500496489612452721895412666, 3.17753155662006701469140011937, 3.44988864651985979669287127779, 5.00434082422532510202118093251, 6.04171334815690603582818986250, 7.00869322316551782393899690131, 7.67658837655474783840426718638, 8.293895677495421321252486721070, 9.531033064910239224700536040234, 10.48830299750378133419785235074
