| L(s) = 1 | + (−2.47 + 0.805i)2-s + (1.68 + 2.32i)3-s + (3.87 − 2.81i)4-s + (−1.39 + 1.74i)5-s + (−6.06 − 4.40i)6-s + 3.88i·7-s + (−4.27 + 5.88i)8-s + (−1.62 + 5.00i)9-s + (2.05 − 5.45i)10-s + (−1.51 − 4.65i)11-s + (13.1 + 4.25i)12-s + (−2.40 − 0.781i)13-s + (−3.12 − 9.62i)14-s + (−6.42 − 0.291i)15-s + (2.89 − 8.92i)16-s + (−0.587 + 0.809i)17-s + ⋯ |
| L(s) = 1 | + (−1.75 + 0.569i)2-s + (0.975 + 1.34i)3-s + (1.93 − 1.40i)4-s + (−0.623 + 0.781i)5-s + (−2.47 − 1.79i)6-s + 1.46i·7-s + (−1.51 + 2.08i)8-s + (−0.542 + 1.66i)9-s + (0.648 − 1.72i)10-s + (−0.455 − 1.40i)11-s + (3.78 + 1.22i)12-s + (−0.667 − 0.216i)13-s + (−0.836 − 2.57i)14-s + (−1.65 − 0.0753i)15-s + (0.724 − 2.23i)16-s + (−0.142 + 0.196i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.825 + 0.564i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.825 + 0.564i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.169732 - 0.548411i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.169732 - 0.548411i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (1.39 - 1.74i)T \) |
| 17 | \( 1 + (0.587 - 0.809i)T \) |
| good | 2 | \( 1 + (2.47 - 0.805i)T + (1.61 - 1.17i)T^{2} \) |
| 3 | \( 1 + (-1.68 - 2.32i)T + (-0.927 + 2.85i)T^{2} \) |
| 7 | \( 1 - 3.88iT - 7T^{2} \) |
| 11 | \( 1 + (1.51 + 4.65i)T + (-8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (2.40 + 0.781i)T + (10.5 + 7.64i)T^{2} \) |
| 19 | \( 1 + (-3.92 - 2.85i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (1.61 - 0.523i)T + (18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (0.834 - 0.606i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-7.37 - 5.35i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (1.46 + 0.475i)T + (29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (2.37 - 7.30i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 8.69iT - 43T^{2} \) |
| 47 | \( 1 + (-3.91 - 5.38i)T + (-14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (0.695 + 0.957i)T + (-16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-2.12 + 6.54i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (2.73 + 8.41i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (0.0931 - 0.128i)T + (-20.7 - 63.7i)T^{2} \) |
| 71 | \( 1 + (3.77 - 2.74i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (6.05 - 1.96i)T + (59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (4.70 - 3.41i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (0.722 - 0.994i)T + (-25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (-1.70 - 5.25i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-4.26 - 5.87i)T + (-29.9 + 92.2i)T^{2} \) |
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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
−11.19351206219719944131629520165, −10.36319174076275112328188229151, −9.783912207323724489042837173644, −8.845540413418370688725483647559, −8.348509496279967718156213450127, −7.70716063630427093991224750662, −6.27471490763552207125035098781, −5.26087553914127942517043691759, −3.28817767628330762068620471695, −2.52036062670114551856990002290,
0.56588782907119684289934994102, 1.67069556081974155902100414008, 2.83003002836243009937211367449, 4.34769701613607351642248367622, 6.93555240817839961586914647212, 7.44202323594894080324826077814, 7.76327344585289591737840174790, 8.746045483376336182709586543282, 9.604893595996698697288171893512, 10.31090113749298806619424442989
