| L(s) = 1 | + (−1.05 − 0.190i)2-s + (−1.45 + 0.402i)3-s + (−0.801 − 0.300i)4-s + (1.24 − 3.82i)5-s + (1.61 − 0.144i)6-s + (−1.58 − 2.93i)7-s + (2.62 + 1.56i)8-s + (−0.612 + 0.365i)9-s + (−2.03 + 3.78i)10-s + (−1.24 + 2.90i)11-s + (1.28 + 0.116i)12-s + (−0.588 − 1.37i)13-s + (1.10 + 3.39i)14-s + (−0.272 + 6.06i)15-s + (−1.17 − 1.02i)16-s + (4.75 − 3.45i)17-s + ⋯ |
| L(s) = 1 | + (−0.744 − 0.135i)2-s + (−0.841 + 0.232i)3-s + (−0.400 − 0.150i)4-s + (0.555 − 1.70i)5-s + (0.657 − 0.0591i)6-s + (−0.597 − 1.11i)7-s + (0.927 + 0.553i)8-s + (−0.204 + 0.121i)9-s + (−0.643 + 1.19i)10-s + (−0.374 + 0.875i)11-s + (0.372 + 0.0335i)12-s + (−0.163 − 0.381i)13-s + (0.294 + 0.906i)14-s + (−0.0703 + 1.56i)15-s + (−0.292 − 0.255i)16-s + (1.15 − 0.837i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.303 + 0.952i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.303 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.240304 - 0.328620i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.240304 - 0.328620i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 71 | \( 1 + (-3.48 + 7.67i)T \) |
| good | 2 | \( 1 + (1.05 + 0.190i)T + (1.87 + 0.702i)T^{2} \) |
| 3 | \( 1 + (1.45 - 0.402i)T + (2.57 - 1.53i)T^{2} \) |
| 5 | \( 1 + (-1.24 + 3.82i)T + (-4.04 - 2.93i)T^{2} \) |
| 7 | \( 1 + (1.58 + 2.93i)T + (-3.85 + 5.84i)T^{2} \) |
| 11 | \( 1 + (1.24 - 2.90i)T + (-7.60 - 7.95i)T^{2} \) |
| 13 | \( 1 + (0.588 + 1.37i)T + (-8.98 + 9.39i)T^{2} \) |
| 17 | \( 1 + (-4.75 + 3.45i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (0.0791 + 1.76i)T + (-18.9 + 1.70i)T^{2} \) |
| 23 | \( 1 + (-3.20 - 1.54i)T + (14.3 + 17.9i)T^{2} \) |
| 29 | \( 1 + (-4.42 - 6.69i)T + (-11.3 + 26.6i)T^{2} \) |
| 31 | \( 1 + (1.14 - 0.996i)T + (4.16 - 30.7i)T^{2} \) |
| 37 | \( 1 + (-2.98 + 1.43i)T + (23.0 - 28.9i)T^{2} \) |
| 41 | \( 1 + (-1.81 + 7.93i)T + (-36.9 - 17.7i)T^{2} \) |
| 43 | \( 1 + (0.159 + 1.17i)T + (-41.4 + 11.4i)T^{2} \) |
| 47 | \( 1 + (-0.456 - 0.125i)T + (40.3 + 24.1i)T^{2} \) |
| 53 | \( 1 + (2.63 - 0.990i)T + (39.9 - 34.8i)T^{2} \) |
| 59 | \( 1 + (-2.60 - 0.234i)T + (58.0 + 10.5i)T^{2} \) |
| 61 | \( 1 + (5.39 - 10.0i)T + (-33.6 - 50.9i)T^{2} \) |
| 67 | \( 1 + (1.26 + 0.473i)T + (50.4 + 44.0i)T^{2} \) |
| 73 | \( 1 + (3.33 + 0.605i)T + (68.3 + 25.6i)T^{2} \) |
| 79 | \( 1 + (-5.57 - 3.33i)T + (37.4 + 69.5i)T^{2} \) |
| 83 | \( 1 + (5.12 + 0.461i)T + (81.6 + 14.8i)T^{2} \) |
| 89 | \( 1 + (-7.22 + 2.71i)T + (67.0 - 58.5i)T^{2} \) |
| 97 | \( 1 + (1.78 + 7.83i)T + (-87.3 + 42.0i)T^{2} \) |
| 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
−14.06111332008298375827492269560, −13.15928611257180064721343824118, −12.22984893114523343887469367851, −10.59375432265011234687184196111, −9.856443370134926654680337019250, −8.911503094462232626126420159032, −7.50235268093666284973113930204, −5.39367930623912588474878333351, −4.70628829685908782712885361247, −0.818175336710572210872237878233,
3.06426378338402208826292085238, 5.79493242216075297997189448488, 6.52379631889294515307378104736, 8.075458152917151077677632629483, 9.505735633917513268783396894837, 10.43542816849732443054460510314, 11.48155958308553724178830513700, 12.73175455261744291904715499169, 14.01599533629202319361112953303, 14.99761570550469842332771641888
