| 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
−11.65779914963877990166297856557, −10.14614482032195241761382068705, −9.043851935693741692173251449315, −8.771819991402644404654432067697, −7.57252083284628083672352368694, −6.78792350952020803494344757540, −5.32339403748145841797809381117, −4.36704709865645307560231171161, −3.37524612195166805826533855957, −1.15410342023051251760901810893,
0.854079240984189626153367641145, 3.17433812005596293024219623772, 4.26732822303352957183917251875, 5.19833027999929359224129609737, 6.38940976564418248519059318572, 7.54072687685712261574764655728, 8.373439295198509207502510132520, 9.450010269542313637761826881667, 10.31916397630902451178553474623, 11.19071366721769240348590025595
