| L(s) = 1 | + (0.939 + 0.342i)2-s + (−0.214 − 1.21i)3-s + (0.766 + 0.642i)4-s + (−2.77 + 2.32i)5-s + (0.214 − 1.21i)6-s + (1.61 − 2.80i)7-s + (0.500 + 0.866i)8-s + (1.38 − 0.503i)9-s + (−3.39 + 1.23i)10-s + (1.61 + 2.80i)11-s + (0.618 − 1.07i)12-s + (0.239 − 1.36i)13-s + (2.47 − 2.08i)14-s + (3.42 + 2.87i)15-s + (0.173 + 0.984i)16-s + (3.17 + 1.15i)17-s + ⋯ |
| L(s) = 1 | + (0.664 + 0.241i)2-s + (−0.123 − 0.702i)3-s + (0.383 + 0.321i)4-s + (−1.23 + 1.04i)5-s + (0.0876 − 0.496i)6-s + (0.611 − 1.05i)7-s + (0.176 + 0.306i)8-s + (0.461 − 0.167i)9-s + (−1.07 + 0.391i)10-s + (0.487 + 0.844i)11-s + (0.178 − 0.309i)12-s + (0.0665 − 0.377i)13-s + (0.662 − 0.555i)14-s + (0.884 + 0.742i)15-s + (0.0434 + 0.246i)16-s + (0.770 + 0.280i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0789i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0789i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.06874 + 0.0817778i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.06874 + 0.0817778i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.939 - 0.342i)T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + (0.214 + 1.21i)T + (-2.81 + 1.02i)T^{2} \) |
| 5 | \( 1 + (2.77 - 2.32i)T + (0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (-1.61 + 2.80i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.61 - 2.80i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.239 + 1.36i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (-3.17 - 1.15i)T + (13.0 + 10.9i)T^{2} \) |
| 23 | \( 1 + (-4.01 - 3.36i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-8.54 + 3.10i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (-0.618 + 1.07i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 8.38T + 37T^{2} \) |
| 41 | \( 1 + (-0.148 - 0.841i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (7.07 - 5.93i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (4.20 - 1.52i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (4.66 + 3.91i)T + (9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (-0.443 - 0.161i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-1.05 - 0.888i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (11.0 - 4.00i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-2.25 + 1.89i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (0.975 + 5.53i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (0.479 + 2.72i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-0.236 + 0.408i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-1.53 + 8.71i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (8.09 + 2.94i)T + (74.3 + 62.3i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.62929201634047621782410695809, −9.829114422588322645152479104603, −7.975318482803733517233239305897, −7.69795098844141005000215452184, −6.94871921717804278558578930612, −6.34519445334402802310657327573, −4.68399652625266961274651786930, −4.01834319828471979612379811189, −3.01270786156636152589339082146, −1.25902154061758924167220888043,
1.22285990181433165721710624350, 3.04547317561988738218934204925, 4.13264707632212056300595588076, 4.81264452405169852827023832336, 5.44806398239184922048875696733, 6.77248382244767682270815864555, 8.045187698474205316742173515240, 8.658724750300925813436255528961, 9.468684098215098106637471219300, 10.66814196510403660525422615558
