| L(s) = 1 | + (−0.167 − 1.40i)2-s + (−1.97 − 1.34i)3-s + (−1.94 + 0.471i)4-s + (−0.0159 − 0.00119i)5-s + (−1.56 + 3.00i)6-s + (−2.59 + 0.509i)7-s + (0.987 + 2.65i)8-s + (0.999 + 2.54i)9-s + (0.000997 + 0.0225i)10-s + (1.65 + 0.649i)11-s + (4.48 + 1.69i)12-s + (−2.07 + 1.65i)13-s + (1.15 + 3.56i)14-s + (0.0299 + 0.0238i)15-s + (3.55 − 1.83i)16-s + (−3.29 − 3.55i)17-s + ⋯ |
| L(s) = 1 | + (−0.118 − 0.992i)2-s + (−1.14 − 0.778i)3-s + (−0.971 + 0.235i)4-s + (−0.00713 − 0.000534i)5-s + (−0.637 + 1.22i)6-s + (−0.981 + 0.192i)7-s + (0.349 + 0.937i)8-s + (0.333 + 0.849i)9-s + (0.000315 + 0.00714i)10-s + (0.499 + 0.195i)11-s + (1.29 + 0.487i)12-s + (−0.574 + 0.458i)13-s + (0.307 + 0.951i)14-s + (0.00773 + 0.00616i)15-s + (0.888 − 0.457i)16-s + (−0.799 − 0.861i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.154 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.154 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0512515 + 0.0598973i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0512515 + 0.0598973i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.167 + 1.40i)T \) |
| 7 | \( 1 + (2.59 - 0.509i)T \) |
| good | 3 | \( 1 + (1.97 + 1.34i)T + (1.09 + 2.79i)T^{2} \) |
| 5 | \( 1 + (0.0159 + 0.00119i)T + (4.94 + 0.745i)T^{2} \) |
| 11 | \( 1 + (-1.65 - 0.649i)T + (8.06 + 7.48i)T^{2} \) |
| 13 | \( 1 + (2.07 - 1.65i)T + (2.89 - 12.6i)T^{2} \) |
| 17 | \( 1 + (3.29 + 3.55i)T + (-1.27 + 16.9i)T^{2} \) |
| 19 | \( 1 + (0.682 - 1.18i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3.75 - 4.04i)T + (-1.71 - 22.9i)T^{2} \) |
| 29 | \( 1 + (-0.487 - 2.13i)T + (-26.1 + 12.5i)T^{2} \) |
| 31 | \( 1 + (4.96 + 8.59i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.78 - 0.549i)T + (30.5 + 20.8i)T^{2} \) |
| 41 | \( 1 + (1.78 + 3.71i)T + (-25.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + (-4.01 + 8.34i)T + (-26.8 - 33.6i)T^{2} \) |
| 47 | \( 1 + (9.62 - 1.45i)T + (44.9 - 13.8i)T^{2} \) |
| 53 | \( 1 + (8.11 - 2.50i)T + (43.7 - 29.8i)T^{2} \) |
| 59 | \( 1 + (-0.462 - 6.17i)T + (-58.3 + 8.79i)T^{2} \) |
| 61 | \( 1 + (-2.72 + 8.81i)T + (-50.4 - 34.3i)T^{2} \) |
| 67 | \( 1 + (-4.78 + 2.76i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-13.9 - 3.18i)T + (63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (1.28 - 8.51i)T + (-69.7 - 21.5i)T^{2} \) |
| 79 | \( 1 + (0.651 + 0.375i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.16 + 2.71i)T + (-18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (-1.89 + 0.744i)T + (65.2 - 60.5i)T^{2} \) |
| 97 | \( 1 + 12.1iT - 97T^{2} \) |
| 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
−11.78968750499295888746913204706, −11.19762458453048150160898567400, −9.857678159359645039657577270513, −9.215508342576611526518926131722, −7.57627110915766591825904086813, −6.47952129123934021518746756745, −5.38541222174242879241800589554, −3.89203718517378298923582742864, −2.07587862909019989010759496339, −0.079650395628756524293527427063,
3.83655563345321916858319296925, 4.89340595710004538381780185396, 6.07857751754432201688332812471, 6.64211749510933408838798826721, 8.129401740114455812258760628264, 9.420628947413872603865293033848, 10.12015065902249145816278007160, 11.01548159421197706366578980666, 12.35102001220771188112962975248, 13.15924787150462195394165345513
