L(s) = 1 | + (−0.5 − 0.866i)2-s + 3-s + (−0.499 + 0.866i)4-s − 5-s + (−0.5 − 0.866i)6-s + (−1 + 1.73i)7-s + 0.999·8-s − 2·9-s + (0.5 + 0.866i)10-s + (−0.499 + 0.866i)12-s + (−1 − 1.73i)13-s + 1.99·14-s − 15-s + (−0.5 − 0.866i)16-s + (3.72 + 6.44i)17-s + (1 + 1.73i)18-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + 0.577·3-s + (−0.249 + 0.433i)4-s − 0.447·5-s + (−0.204 − 0.353i)6-s + (−0.377 + 0.654i)7-s + 0.353·8-s − 0.666·9-s + (0.158 + 0.273i)10-s + (−0.144 + 0.249i)12-s + (−0.277 − 0.480i)13-s + 0.534·14-s − 0.258·15-s + (−0.125 − 0.216i)16-s + (0.902 + 1.56i)17-s + (0.235 + 0.408i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.376 - 0.926i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.376 - 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.733179 + 0.493293i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.733179 + 0.493293i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 5 | \( 1 + T \) |
| 67 | \( 1 + (7.22 + 3.85i)T \) |
good | 3 | \( 1 - T + 3T^{2} \) |
| 7 | \( 1 + (1 - 1.73i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1 + 1.73i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-3.72 - 6.44i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.72 - 4.71i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3 - 5.19i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (3.22 - 5.58i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.22 - 7.31i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.22 + 3.85i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + 11.4T + 43T^{2} \) |
| 47 | \( 1 + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 4.44T + 53T^{2} \) |
| 59 | \( 1 - 13.4T + 59T^{2} \) |
| 61 | \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \) |
| 71 | \( 1 + (-2.22 + 3.85i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (7.72 + 13.3i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-5.22 - 9.04i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 11.8T + 89T^{2} \) |
| 97 | \( 1 + (4.72 + 8.18i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.55104826064275733993514611656, −9.838035471385218733665744837861, −8.878875930514517102261485795205, −8.248508465070464187111799673429, −7.58232544266138531500585877531, −6.13985326694211103858299743254, −5.20211395523975798788328468998, −3.57227165549340172204953883704, −3.14111112712581098015872382262, −1.69340939986965152604854819180,
0.49211392820243091364537608345, 2.59723595317743128717502856547, 3.72325213058338458500796496860, 4.90033498893668724580348561983, 5.94345582787486563911664055062, 7.28980335435967299213476191099, 7.45970636448954167972164566535, 8.609136223477488161843731237524, 9.431017472622257372641886019936, 9.942017010007376272007253990392