| L(s) = 1 | + (−0.248 − 0.0808i)2-s + (−0.408 + 0.562i)3-s + (−1.56 − 1.13i)4-s + (−1.61 + 1.54i)5-s + (0.147 − 0.106i)6-s − 1.76i·7-s + (0.604 + 0.832i)8-s + (0.777 + 2.39i)9-s + (0.527 − 0.253i)10-s + (1.20 − 3.69i)11-s + (1.27 − 0.414i)12-s + (2.08 − 0.676i)13-s + (−0.143 + 0.440i)14-s + (−0.207 − 1.53i)15-s + (1.11 + 3.41i)16-s + (−0.587 − 0.809i)17-s + ⋯ |
| L(s) = 1 | + (−0.175 − 0.0571i)2-s + (−0.235 + 0.324i)3-s + (−0.781 − 0.567i)4-s + (−0.723 + 0.690i)5-s + (0.0600 − 0.0436i)6-s − 0.668i·7-s + (0.213 + 0.294i)8-s + (0.259 + 0.798i)9-s + (0.166 − 0.0801i)10-s + (0.362 − 1.11i)11-s + (0.368 − 0.119i)12-s + (0.577 − 0.187i)13-s + (−0.0382 + 0.117i)14-s + (−0.0535 − 0.397i)15-s + (0.277 + 0.854i)16-s + (−0.142 − 0.196i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 + 0.410i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.911 + 0.410i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.871121 - 0.187100i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.871121 - 0.187100i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (1.61 - 1.54i)T \) |
| 17 | \( 1 + (0.587 + 0.809i)T \) |
| good | 2 | \( 1 + (0.248 + 0.0808i)T + (1.61 + 1.17i)T^{2} \) |
| 3 | \( 1 + (0.408 - 0.562i)T + (-0.927 - 2.85i)T^{2} \) |
| 7 | \( 1 + 1.76iT - 7T^{2} \) |
| 11 | \( 1 + (-1.20 + 3.69i)T + (-8.89 - 6.46i)T^{2} \) |
| 13 | \( 1 + (-2.08 + 0.676i)T + (10.5 - 7.64i)T^{2} \) |
| 19 | \( 1 + (-4.31 + 3.13i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (-7.61 - 2.47i)T + (18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (1.72 + 1.25i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-8.32 + 6.05i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (3.86 - 1.25i)T + (29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-2.36 - 7.28i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 4.18iT - 43T^{2} \) |
| 47 | \( 1 + (5.92 - 8.16i)T + (-14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (-1.99 + 2.74i)T + (-16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (2.93 + 9.04i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-1.67 + 5.15i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + (-5.25 - 7.22i)T + (-20.7 + 63.7i)T^{2} \) |
| 71 | \( 1 + (-0.990 - 0.719i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (8.89 + 2.89i)T + (59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (7.05 + 5.12i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-9.22 - 12.6i)T + (-25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (-2.92 + 9.00i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (0.0568 - 0.0782i)T + (-29.9 - 92.2i)T^{2} \) |
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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
−11.19071366721769240348590025595, −10.31916397630902451178553474623, −9.450010269542313637761826881667, −8.373439295198509207502510132520, −7.54072687685712261574764655728, −6.38940976564418248519059318572, −5.19833027999929359224129609737, −4.26732822303352957183917251875, −3.17433812005596293024219623772, −0.854079240984189626153367641145,
1.15410342023051251760901810893, 3.37524612195166805826533855957, 4.36704709865645307560231171161, 5.32339403748145841797809381117, 6.78792350952020803494344757540, 7.57252083284628083672352368694, 8.771819991402644404654432067697, 9.043851935693741692173251449315, 10.14614482032195241761382068705, 11.65779914963877990166297856557
