L(s) = 1 | + (−2.02 + 0.925i)2-s + (−0.281 − 0.959i)3-s + (1.94 − 2.24i)4-s + (1.45 + 1.68i)6-s + (0.0872 − 0.0125i)7-s + (−0.607 + 2.06i)8-s + (1.68 − 1.08i)9-s + (1.01 − 2.21i)11-s + (−2.69 − 1.23i)12-s + (−3.56 − 0.512i)13-s + (−0.165 + 0.106i)14-s + (0.160 + 1.11i)16-s + (2.55 − 2.21i)17-s + (−2.40 + 3.74i)18-s + (−1.87 + 2.16i)19-s + ⋯ |
L(s) = 1 | + (−1.43 + 0.654i)2-s + (−0.162 − 0.553i)3-s + (0.971 − 1.12i)4-s + (0.595 + 0.687i)6-s + (0.0329 − 0.00474i)7-s + (−0.214 + 0.731i)8-s + (0.560 − 0.360i)9-s + (0.305 − 0.669i)11-s + (−0.779 − 0.355i)12-s + (−0.988 − 0.142i)13-s + (−0.0441 + 0.0283i)14-s + (0.0402 + 0.279i)16-s + (0.620 − 0.537i)17-s + (−0.568 + 0.883i)18-s + (−0.429 + 0.496i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.120 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.120 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.389237 - 0.344832i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.389237 - 0.344832i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 23 | \( 1 + (-2.15 - 4.28i)T \) |
good | 2 | \( 1 + (2.02 - 0.925i)T + (1.30 - 1.51i)T^{2} \) |
| 3 | \( 1 + (0.281 + 0.959i)T + (-2.52 + 1.62i)T^{2} \) |
| 7 | \( 1 + (-0.0872 + 0.0125i)T + (6.71 - 1.97i)T^{2} \) |
| 11 | \( 1 + (-1.01 + 2.21i)T + (-7.20 - 8.31i)T^{2} \) |
| 13 | \( 1 + (3.56 + 0.512i)T + (12.4 + 3.66i)T^{2} \) |
| 17 | \( 1 + (-2.55 + 2.21i)T + (2.41 - 16.8i)T^{2} \) |
| 19 | \( 1 + (1.87 - 2.16i)T + (-2.70 - 18.8i)T^{2} \) |
| 29 | \( 1 + (4.87 + 5.62i)T + (-4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (-1.65 - 0.485i)T + (26.0 + 16.7i)T^{2} \) |
| 37 | \( 1 + (-0.382 - 0.595i)T + (-15.3 + 33.6i)T^{2} \) |
| 41 | \( 1 + (4.66 + 3.00i)T + (17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (0.675 + 2.30i)T + (-36.1 + 23.2i)T^{2} \) |
| 47 | \( 1 + 11.6iT - 47T^{2} \) |
| 53 | \( 1 + (-10.2 + 1.47i)T + (50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (-1.64 + 11.4i)T + (-56.6 - 16.6i)T^{2} \) |
| 61 | \( 1 + (3.89 + 1.14i)T + (51.3 + 32.9i)T^{2} \) |
| 67 | \( 1 + (-11.1 + 5.07i)T + (43.8 - 50.6i)T^{2} \) |
| 71 | \( 1 + (-0.787 - 1.72i)T + (-46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (4.16 + 3.61i)T + (10.3 + 72.2i)T^{2} \) |
| 79 | \( 1 + (-0.997 + 6.93i)T + (-75.7 - 22.2i)T^{2} \) |
| 83 | \( 1 + (2.25 + 3.51i)T + (-34.4 + 75.4i)T^{2} \) |
| 89 | \( 1 + (15.3 - 4.49i)T + (74.8 - 48.1i)T^{2} \) |
| 97 | \( 1 + (7.77 - 12.0i)T + (-40.2 - 88.2i)T^{2} \) |
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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
−10.03848396316697324418465829748, −9.686324286176476641862418973839, −8.670377490306049075809490587863, −7.78223813419543063137929692362, −7.17345963533746745013541490867, −6.38839054686969701421060698470, −5.36738339023161694636176292638, −3.69700244764505996029891056590, −1.85048901650731965428755662109, −0.50532733697076014473038353560,
1.49909094777513155010171442308, 2.70045468147768562505291248993, 4.24546120510149155018624425470, 5.20794816715481955101010520459, 6.86676244567740513867063208276, 7.56358118242801264807883819419, 8.559547881162993319385661602014, 9.443251419928465471384620069539, 10.00667269399527044793969839467, 10.64426112315596014580799491029