L(s) = 1 | + (0.164 + 1.40i)2-s + (1.79 − 1.03i)3-s + (−1.94 + 0.462i)4-s − 5-s + (1.75 + 2.35i)6-s + (3.65 + 2.10i)7-s + (−0.969 − 2.65i)8-s + (0.658 − 1.14i)9-s + (−0.164 − 1.40i)10-s + (2.34 + 4.05i)11-s + (−3.02 + 2.85i)12-s + (−3.60 − 0.189i)13-s + (−2.36 + 5.48i)14-s + (−1.79 + 1.03i)15-s + (3.57 − 1.79i)16-s + (1.44 − 2.49i)17-s + ⋯ |
L(s) = 1 | + (0.116 + 0.993i)2-s + (1.03 − 0.599i)3-s + (−0.972 + 0.231i)4-s − 0.447·5-s + (0.716 + 0.961i)6-s + (1.38 + 0.797i)7-s + (−0.342 − 0.939i)8-s + (0.219 − 0.380i)9-s + (−0.0520 − 0.444i)10-s + (0.705 + 1.22i)11-s + (−0.872 + 0.823i)12-s + (−0.998 − 0.0524i)13-s + (−0.631 + 1.46i)14-s + (−0.464 + 0.268i)15-s + (0.893 − 0.449i)16-s + (0.349 − 0.605i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.236 - 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.236 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.58069 + 1.24169i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.58069 + 1.24169i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.164 - 1.40i)T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + (3.60 + 0.189i)T \) |
good | 3 | \( 1 + (-1.79 + 1.03i)T + (1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 + (-3.65 - 2.10i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.34 - 4.05i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-1.44 + 2.49i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.09 + 5.36i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.36 - 5.82i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (4.64 - 2.68i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 0.540iT - 31T^{2} \) |
| 37 | \( 1 + (-0.815 - 1.41i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.36 + 0.789i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (4.99 + 2.88i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 4.24iT - 47T^{2} \) |
| 53 | \( 1 + 10.7iT - 53T^{2} \) |
| 59 | \( 1 + (-6.90 + 11.9i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (6.45 + 3.72i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (5.52 + 9.57i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (8.93 + 5.15i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 8.13iT - 73T^{2} \) |
| 79 | \( 1 - 1.61T + 79T^{2} \) |
| 83 | \( 1 + 3.34T + 83T^{2} \) |
| 89 | \( 1 + (2.76 - 1.59i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (12.2 + 7.09i)T + (48.5 + 84.0i)T^{2} \) |
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
−11.34812029996598968280292939024, −9.520449079200257559271561267291, −9.128312819851898963906718324157, −8.125426306612772650557729042919, −7.43653995082466444645552968475, −7.03685339576331675369178819396, −5.21862668094379599661123532025, −4.76626397580538026480148602963, −3.19450128928234869027592457659, −1.79368889650067509760944600728,
1.24682084709249682841399040398, 2.81105654334428395697052199666, 3.87137323456400241489416727013, 4.39284365101256594036722447381, 5.66834811119193258009147688515, 7.53835656986924000920804129674, 8.325656035545938346213757016640, 8.878885258517545813702322271973, 9.963068225141115831296842798094, 10.63915080192071895087921246010