| L(s) = 1 | + (0.536 + 1.30i)2-s + (−0.540 − 0.841i)3-s + (−1.42 + 1.40i)4-s + (−2.26 + 1.03i)5-s + (0.810 − 1.15i)6-s + (−2.49 + 0.732i)7-s + (−2.60 − 1.11i)8-s + (−0.415 + 0.909i)9-s + (−2.57 − 2.41i)10-s + (3.71 − 3.21i)11-s + (1.95 + 0.439i)12-s + (1.81 − 6.18i)13-s + (−2.29 − 2.87i)14-s + (2.09 + 1.34i)15-s + (0.0605 − 3.99i)16-s + (0.253 − 1.76i)17-s + ⋯ |
| L(s) = 1 | + (0.379 + 0.925i)2-s + (−0.312 − 0.485i)3-s + (−0.712 + 0.701i)4-s + (−1.01 + 0.463i)5-s + (0.331 − 0.472i)6-s + (−0.943 + 0.276i)7-s + (−0.919 − 0.393i)8-s + (−0.138 + 0.303i)9-s + (−0.813 − 0.763i)10-s + (1.12 − 0.970i)11-s + (0.563 + 0.126i)12-s + (0.503 − 1.71i)13-s + (−0.613 − 0.767i)14-s + (0.541 + 0.348i)15-s + (0.0151 − 0.999i)16-s + (0.0614 − 0.427i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.529 + 0.848i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.529 + 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.533325 - 0.295796i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.533325 - 0.295796i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.536 - 1.30i)T \) |
| 3 | \( 1 + (0.540 + 0.841i)T \) |
| 23 | \( 1 + (-1.87 - 4.41i)T \) |
| good | 5 | \( 1 + (2.26 - 1.03i)T + (3.27 - 3.77i)T^{2} \) |
| 7 | \( 1 + (2.49 - 0.732i)T + (5.88 - 3.78i)T^{2} \) |
| 11 | \( 1 + (-3.71 + 3.21i)T + (1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-1.81 + 6.18i)T + (-10.9 - 7.02i)T^{2} \) |
| 17 | \( 1 + (-0.253 + 1.76i)T + (-16.3 - 4.78i)T^{2} \) |
| 19 | \( 1 + (-2.93 + 0.422i)T + (18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (8.21 + 1.18i)T + (27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (7.54 + 4.85i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (6.44 + 2.94i)T + (24.2 + 27.9i)T^{2} \) |
| 41 | \( 1 + (1.79 + 3.93i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (2.68 + 4.17i)T + (-17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 - 8.53T + 47T^{2} \) |
| 53 | \( 1 + (1.11 + 3.78i)T + (-44.5 + 28.6i)T^{2} \) |
| 59 | \( 1 + (4.07 - 13.8i)T + (-49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (-5.74 + 8.94i)T + (-25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (7.32 + 6.34i)T + (9.53 + 66.3i)T^{2} \) |
| 71 | \( 1 + (3.72 - 4.30i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (1.02 + 7.12i)T + (-70.0 + 20.5i)T^{2} \) |
| 79 | \( 1 + (-0.901 - 0.264i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (-5.74 - 2.62i)T + (54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (1.07 - 0.688i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (3.10 + 6.80i)T + (-63.5 + 73.3i)T^{2} \) |
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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
−10.90837602616693731099842345265, −9.461353821667558035610893840571, −8.684889776235289123894718423741, −7.54925658088708712609765932981, −7.19056069809219635452907094965, −5.94329142367241832177080646018, −5.52709950106848916002990812933, −3.58378723064357754641292622446, −3.38148879216944603854726102198, −0.33943916225357338842658724485,
1.56447838244810025725539941350, 3.56321839833765390602963157223, 4.02799944832507003490356326394, 4.88315317129959007615065223521, 6.29823711368860431526097977682, 7.13494924115909879745999119155, 8.825634462301294799203642741611, 9.248716981551833414566030180452, 10.11175581696405599706119525351, 11.13626587278504169887243839239
