L(s) = 1 | + (1.23 + 2.14i)3-s + (1.06 − 1.83i)5-s + (−2.63 + 0.272i)7-s + (−1.56 + 2.70i)9-s + (2.39 + 4.14i)11-s + 13-s + 5.25·15-s + (1.88 + 3.27i)17-s + (−1.78 + 3.08i)19-s + (−3.83 − 5.30i)21-s + (2.23 − 3.87i)23-s + (0.246 + 0.427i)25-s − 0.303·27-s − 5.90·29-s + (−1.88 − 3.26i)31-s + ⋯ |
L(s) = 1 | + (0.714 + 1.23i)3-s + (0.474 − 0.822i)5-s + (−0.994 + 0.102i)7-s + (−0.520 + 0.901i)9-s + (0.721 + 1.25i)11-s + 0.277·13-s + 1.35·15-s + (0.458 + 0.793i)17-s + (−0.409 + 0.708i)19-s + (−0.837 − 1.15i)21-s + (0.466 − 0.807i)23-s + (0.0493 + 0.0855i)25-s − 0.0584·27-s − 1.09·29-s + (−0.338 − 0.586i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.165 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.165 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.077840967\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.077840967\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (2.63 - 0.272i)T \) |
| 13 | \( 1 - T \) |
good | 3 | \( 1 + (-1.23 - 2.14i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (-1.06 + 1.83i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.39 - 4.14i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-1.88 - 3.27i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.78 - 3.08i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.23 + 3.87i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 5.90T + 29T^{2} \) |
| 31 | \( 1 + (1.88 + 3.26i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.81 - 4.87i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 10.3T + 41T^{2} \) |
| 43 | \( 1 + 3.40T + 43T^{2} \) |
| 47 | \( 1 + (3.55 - 6.15i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-6.19 - 10.7i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.39 - 4.14i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (1.60 - 2.77i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.44 + 2.51i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 2.53T + 71T^{2} \) |
| 73 | \( 1 + (3.85 + 6.66i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (2.58 - 4.48i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 3.46T + 83T^{2} \) |
| 89 | \( 1 + (1.83 - 3.17i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 5.40T + 97T^{2} \) |
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
−9.644033502489734630956479086616, −9.119475289585558120689010654270, −8.547565614519080563896973231063, −7.41582602060143716710102142701, −6.32798907204166739103025103924, −5.51754396783819158568287294390, −4.38711991095515347815732278219, −3.95464454188743556836929296556, −2.87400493448617184239428617983, −1.57417528781564671298395823363,
0.78386515899032747926816796072, 2.14819976081084681209895628922, 3.07180162983167566621298678053, 3.64264533303916724942465084664, 5.48773280249254808505146353800, 6.31749856519704235387285510368, 6.91675948551493455021855788519, 7.43097654908803567542669244032, 8.580674056123888321627245993912, 9.116032722970558226865563076139