| L(s) = 1 | + (0.535 − 0.844i)2-s + (1.28 − 1.20i)3-s + (−0.425 − 0.904i)4-s + (−2.15 − 0.582i)5-s + (−0.331 − 1.73i)6-s + (3.35 − 2.43i)7-s + (−0.992 − 0.125i)8-s + (0.00769 − 0.122i)9-s + (−1.64 + 1.51i)10-s + (−2.53 + 4.00i)11-s + (−1.64 − 0.650i)12-s + (0.197 − 3.14i)13-s + (−0.260 − 4.14i)14-s + (−3.48 + 1.86i)15-s + (−0.637 + 0.770i)16-s + (1.11 − 2.36i)17-s + ⋯ |
| L(s) = 1 | + (0.378 − 0.597i)2-s + (0.743 − 0.698i)3-s + (−0.212 − 0.452i)4-s + (−0.965 − 0.260i)5-s + (−0.135 − 0.708i)6-s + (1.26 − 0.921i)7-s + (−0.350 − 0.0443i)8-s + (0.00256 − 0.0407i)9-s + (−0.521 + 0.477i)10-s + (−0.765 + 1.20i)11-s + (−0.474 − 0.187i)12-s + (0.0548 − 0.871i)13-s + (−0.0696 − 1.10i)14-s + (−0.900 + 0.480i)15-s + (−0.159 + 0.192i)16-s + (0.269 − 0.572i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.272 + 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.272 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.02537 - 1.35609i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.02537 - 1.35609i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.535 + 0.844i)T \) |
| 5 | \( 1 + (2.15 + 0.582i)T \) |
| good | 3 | \( 1 + (-1.28 + 1.20i)T + (0.188 - 2.99i)T^{2} \) |
| 7 | \( 1 + (-3.35 + 2.43i)T + (2.16 - 6.65i)T^{2} \) |
| 11 | \( 1 + (2.53 - 4.00i)T + (-4.68 - 9.95i)T^{2} \) |
| 13 | \( 1 + (-0.197 + 3.14i)T + (-12.8 - 1.62i)T^{2} \) |
| 17 | \( 1 + (-1.11 + 2.36i)T + (-10.8 - 13.0i)T^{2} \) |
| 19 | \( 1 + (-2.31 - 2.17i)T + (1.19 + 18.9i)T^{2} \) |
| 23 | \( 1 + (3.93 + 1.00i)T + (20.1 + 11.0i)T^{2} \) |
| 29 | \( 1 + (-7.05 - 3.88i)T + (15.5 + 24.4i)T^{2} \) |
| 31 | \( 1 + (0.0815 - 0.173i)T + (-19.7 - 23.8i)T^{2} \) |
| 37 | \( 1 + (-0.193 + 0.234i)T + (-6.93 - 36.3i)T^{2} \) |
| 41 | \( 1 + (5.60 - 1.44i)T + (35.9 - 19.7i)T^{2} \) |
| 43 | \( 1 + (-0.158 + 0.487i)T + (-34.7 - 25.2i)T^{2} \) |
| 47 | \( 1 + (-13.1 + 1.66i)T + (45.5 - 11.6i)T^{2} \) |
| 53 | \( 1 + (2.32 - 12.1i)T + (-49.2 - 19.5i)T^{2} \) |
| 59 | \( 1 + (8.20 + 3.25i)T + (43.0 + 40.3i)T^{2} \) |
| 61 | \( 1 + (4.63 + 1.19i)T + (53.4 + 29.3i)T^{2} \) |
| 67 | \( 1 + (3.34 - 1.83i)T + (35.9 - 56.5i)T^{2} \) |
| 71 | \( 1 + (-6.03 + 0.763i)T + (68.7 - 17.6i)T^{2} \) |
| 73 | \( 1 + (12.2 - 4.83i)T + (53.2 - 49.9i)T^{2} \) |
| 79 | \( 1 + (7.46 - 7.01i)T + (4.96 - 78.8i)T^{2} \) |
| 83 | \( 1 + (4.03 + 3.79i)T + (5.21 + 82.8i)T^{2} \) |
| 89 | \( 1 + (-10.8 + 4.31i)T + (64.8 - 60.9i)T^{2} \) |
| 97 | \( 1 + (-4.71 - 2.59i)T + (51.9 + 81.8i)T^{2} \) |
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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
−12.05767830847084946428059595955, −10.84165722353644868149960970953, −10.19542723282117313943066618973, −8.569753386312637257748094888001, −7.71462668286730919580288278921, −7.34646060707874663995281664902, −5.13824296521157688758247918217, −4.33661257253732867793260103777, −2.87039038997782809233258350411, −1.36505862754557274874639262708,
2.78942986813186061580544275783, 3.94883817720736727454893313435, 4.95205945904558459346696017615, 6.18878297180724666360737615971, 7.74349449450284004690745485886, 8.386064586601700573255131776283, 8.991857665927645050194587636005, 10.49860601063334307693629929939, 11.60917526017138737797807711503, 12.08594713292855685265282143550
