| L(s) = 1 | + (0.868 − 1.28i)2-s + (0.298 + 1.70i)3-s + (−0.146 − 0.368i)4-s + (−0.104 + 0.224i)5-s + (2.44 + 1.09i)6-s + (0.721 + 2.59i)7-s + (2.42 + 0.533i)8-s + (−2.82 + 1.01i)9-s + (0.197 + 0.328i)10-s + (−3.37 − 3.19i)11-s + (0.585 − 0.360i)12-s + (0.511 − 4.70i)13-s + (3.95 + 1.33i)14-s + (−0.414 − 0.110i)15-s + (3.36 − 3.18i)16-s + (5.31 + 1.47i)17-s + ⋯ |
| L(s) = 1 | + (0.614 − 0.906i)2-s + (0.172 + 0.985i)3-s + (−0.0734 − 0.184i)4-s + (−0.0465 + 0.100i)5-s + (0.998 + 0.448i)6-s + (0.272 + 0.981i)7-s + (0.856 + 0.188i)8-s + (−0.940 + 0.339i)9-s + (0.0625 + 0.103i)10-s + (−1.01 − 0.963i)11-s + (0.168 − 0.104i)12-s + (0.142 − 1.30i)13-s + (1.05 + 0.356i)14-s + (−0.107 − 0.0284i)15-s + (0.841 − 0.797i)16-s + (1.29 + 0.358i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0719i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0719i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.66815 + 0.0600497i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.66815 + 0.0600497i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.298 - 1.70i)T \) |
| 59 | \( 1 + (6.24 - 4.47i)T \) |
| good | 2 | \( 1 + (-0.868 + 1.28i)T + (-0.740 - 1.85i)T^{2} \) |
| 5 | \( 1 + (0.104 - 0.224i)T + (-3.23 - 3.81i)T^{2} \) |
| 7 | \( 1 + (-0.721 - 2.59i)T + (-5.99 + 3.60i)T^{2} \) |
| 11 | \( 1 + (3.37 + 3.19i)T + (0.595 + 10.9i)T^{2} \) |
| 13 | \( 1 + (-0.511 + 4.70i)T + (-12.6 - 2.79i)T^{2} \) |
| 17 | \( 1 + (-5.31 - 1.47i)T + (14.5 + 8.76i)T^{2} \) |
| 19 | \( 1 + (2.81 + 2.14i)T + (5.08 + 18.3i)T^{2} \) |
| 23 | \( 1 + (3.69 + 6.96i)T + (-12.9 + 19.0i)T^{2} \) |
| 29 | \( 1 + (3.87 - 2.63i)T + (10.7 - 26.9i)T^{2} \) |
| 31 | \( 1 + (-0.174 - 0.229i)T + (-8.29 + 29.8i)T^{2} \) |
| 37 | \( 1 + (-2.05 - 9.34i)T + (-33.5 + 15.5i)T^{2} \) |
| 41 | \( 1 + (6.09 + 3.23i)T + (23.0 + 33.9i)T^{2} \) |
| 43 | \( 1 + (1.05 + 1.11i)T + (-2.32 + 42.9i)T^{2} \) |
| 47 | \( 1 + (0.0458 - 0.0212i)T + (30.4 - 35.8i)T^{2} \) |
| 53 | \( 1 + (-1.21 + 2.01i)T + (-24.8 - 46.8i)T^{2} \) |
| 61 | \( 1 + (0.518 + 0.351i)T + (22.5 + 56.6i)T^{2} \) |
| 67 | \( 1 + (1.33 - 6.05i)T + (-60.8 - 28.1i)T^{2} \) |
| 71 | \( 1 + (-3.81 - 8.23i)T + (-45.9 + 54.1i)T^{2} \) |
| 73 | \( 1 + (-0.196 + 0.583i)T + (-58.1 - 44.1i)T^{2} \) |
| 79 | \( 1 + (-0.518 + 9.55i)T + (-78.5 - 8.54i)T^{2} \) |
| 83 | \( 1 + (-1.54 + 9.40i)T + (-78.6 - 26.5i)T^{2} \) |
| 89 | \( 1 + (-7.00 - 10.3i)T + (-32.9 + 82.6i)T^{2} \) |
| 97 | \( 1 + (3.08 + 9.16i)T + (-77.2 + 58.7i)T^{2} \) |
| 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
−12.62486184401919588453158412541, −11.67557687573791209190779844420, −10.66732284488221691291145100791, −10.27303349666850983885823590889, −8.608213015707791037770135070079, −7.999866846227750596762201926393, −5.71783676493892789146798982501, −4.96763118545885837854810425311, −3.41304962433667510623779122004, −2.66212840870750245405437355881,
1.75780725614958219978101329960, 4.03861458150675804851504164735, 5.32288226894675846413547816347, 6.48693177982396856518792242439, 7.49356914793123594439047370214, 7.86924519228038392724347771229, 9.649691332827214171327480962098, 10.77805115310563548258731918100, 12.02569331700915403903818335170, 12.97854638096894616198750363780
