L(s) = 1 | + (0.618 − 1.07i)3-s + (−1.61 − 2.80i)5-s + (0.736 + 1.27i)9-s + (−3.23 + 5.60i)11-s + 0.763·13-s − 4·15-s + (−2.23 + 3.87i)17-s + (−0.618 − 1.07i)19-s + (2 + 3.46i)23-s + (−2.73 + 4.73i)25-s + 5.52·27-s − 4.47·29-s + (−1.23 + 2.14i)31-s + (3.99 + 6.92i)33-s + (2.23 + 3.87i)37-s + ⋯ |
L(s) = 1 | + (0.356 − 0.618i)3-s + (−0.723 − 1.25i)5-s + (0.245 + 0.424i)9-s + (−0.975 + 1.68i)11-s + 0.211·13-s − 1.03·15-s + (−0.542 + 0.939i)17-s + (−0.141 − 0.245i)19-s + (0.417 + 0.722i)23-s + (−0.547 + 0.947i)25-s + 1.06·27-s − 0.830·29-s + (−0.222 + 0.384i)31-s + (0.696 + 1.20i)33-s + (0.367 + 0.636i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9283799455\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9283799455\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-0.618 + 1.07i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (1.61 + 2.80i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (3.23 - 5.60i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 0.763T + 13T^{2} \) |
| 17 | \( 1 + (2.23 - 3.87i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.618 + 1.07i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2 - 3.46i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 4.47T + 29T^{2} \) |
| 31 | \( 1 + (1.23 - 2.14i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.23 - 3.87i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 8.47T + 41T^{2} \) |
| 43 | \( 1 + 6.47T + 43T^{2} \) |
| 47 | \( 1 + (-5.23 - 9.06i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-5 + 8.66i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (4.61 - 7.99i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (5.61 + 9.73i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 4.94T + 71T^{2} \) |
| 73 | \( 1 + (-1.47 + 2.54i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.47 - 11.2i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 9.23T + 83T^{2} \) |
| 89 | \( 1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 12.4T + 97T^{2} \) |
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
−9.433097241075489770499640339518, −8.598345758658559653893532902415, −7.924841846604238926411104750192, −7.45788556561610143789598912212, −6.56227331910042142521843247123, −5.08219931586009969905089441816, −4.78109764087852007924899706430, −3.72614115959193451790583610042, −2.24093177878760693690444564115, −1.42797715517561468564726213099,
0.34602097072526359280066322291, 2.57289404709095337168354751736, 3.30466200013414942013579493394, 3.88617041770536397016379778328, 5.07844481449070923253486601402, 6.12472487427789030911784089386, 6.91917392386609494132689691239, 7.69435000150578282821794383907, 8.561542953801290089233846946743, 9.182001448623294826133923782041