| L(s) = 1 | + (1.32 + 1.60i)2-s + (2.43 − 0.307i)3-s + (−0.434 + 2.28i)4-s + (3.72 + 3.49i)6-s + (−0.629 + 0.457i)7-s + (−0.584 + 0.321i)8-s + (2.93 − 0.753i)9-s + (2.06 + 2.49i)11-s + (−0.357 + 5.68i)12-s + (−3.07 + 0.789i)13-s + (−1.56 − 0.402i)14-s + (3.02 + 1.19i)16-s + (−1.16 − 6.10i)17-s + (5.09 + 3.70i)18-s + (−4.38 − 0.553i)19-s + ⋯ |
| L(s) = 1 | + (0.936 + 1.13i)2-s + (1.40 − 0.177i)3-s + (−0.217 + 1.14i)4-s + (1.51 + 1.42i)6-s + (−0.237 + 0.172i)7-s + (−0.206 + 0.113i)8-s + (0.977 − 0.251i)9-s + (0.621 + 0.750i)11-s + (−0.103 + 1.64i)12-s + (−0.852 + 0.218i)13-s + (−0.418 − 0.107i)14-s + (0.756 + 0.299i)16-s + (−0.282 − 1.48i)17-s + (1.20 + 0.872i)18-s + (−1.00 − 0.127i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.179 - 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.179 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.77405 + 2.31298i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.77405 + 2.31298i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 + (-1.32 - 1.60i)T + (-0.374 + 1.96i)T^{2} \) |
| 3 | \( 1 + (-2.43 + 0.307i)T + (2.90 - 0.746i)T^{2} \) |
| 7 | \( 1 + (0.629 - 0.457i)T + (2.16 - 6.65i)T^{2} \) |
| 11 | \( 1 + (-2.06 - 2.49i)T + (-2.06 + 10.8i)T^{2} \) |
| 13 | \( 1 + (3.07 - 0.789i)T + (11.3 - 6.26i)T^{2} \) |
| 17 | \( 1 + (1.16 + 6.10i)T + (-15.8 + 6.25i)T^{2} \) |
| 19 | \( 1 + (4.38 + 0.553i)T + (18.4 + 4.72i)T^{2} \) |
| 23 | \( 1 + (-2.87 + 4.53i)T + (-9.79 - 20.8i)T^{2} \) |
| 29 | \( 1 + (-2.56 - 5.44i)T + (-18.4 + 22.3i)T^{2} \) |
| 31 | \( 1 + (-0.00238 - 0.0125i)T + (-28.8 + 11.4i)T^{2} \) |
| 37 | \( 1 + (6.84 + 2.70i)T + (26.9 + 25.3i)T^{2} \) |
| 41 | \( 1 + (6.20 + 9.77i)T + (-17.4 + 37.0i)T^{2} \) |
| 43 | \( 1 + (1.76 - 5.43i)T + (-34.7 - 25.2i)T^{2} \) |
| 47 | \( 1 + (-5.86 - 3.22i)T + (25.1 + 39.6i)T^{2} \) |
| 53 | \( 1 + (1.75 - 1.64i)T + (3.32 - 52.8i)T^{2} \) |
| 59 | \( 1 + (-0.0986 + 1.56i)T + (-58.5 - 7.39i)T^{2} \) |
| 61 | \( 1 + (-7.16 + 11.2i)T + (-25.9 - 55.1i)T^{2} \) |
| 67 | \( 1 + (-3.96 + 8.43i)T + (-42.7 - 51.6i)T^{2} \) |
| 71 | \( 1 + (-4.47 - 2.45i)T + (38.0 + 59.9i)T^{2} \) |
| 73 | \( 1 + (0.634 + 10.0i)T + (-72.4 + 9.14i)T^{2} \) |
| 79 | \( 1 + (-1.81 + 0.229i)T + (76.5 - 19.6i)T^{2} \) |
| 83 | \( 1 + (3.86 + 0.488i)T + (80.3 + 20.6i)T^{2} \) |
| 89 | \( 1 + (-0.254 - 4.04i)T + (-88.2 + 11.1i)T^{2} \) |
| 97 | \( 1 + (-0.777 - 1.65i)T + (-61.8 + 74.7i)T^{2} \) |
| 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
−10.68170918658145578852762566415, −9.497575074514526447959676297522, −8.915694802215476750969953285671, −7.957814323585520185444864406967, −6.97448196999572752217925220060, −6.74043025301924717035344688112, −5.13987634833950387431508322062, −4.41485989734722843231040918415, −3.27481287405886534406720184447, −2.16740430247146975378633568969,
1.71661035052158138430034589799, 2.74539570233518502098139472275, 3.63482585735395481996504540097, 4.20768996025378312929994534106, 5.53030684466501806642125701767, 6.81281055656036997762377911312, 8.133497190918332864587345478125, 8.663490014910315095357852267270, 9.834978265499993476105067885806, 10.35712435711080468020001289558
