L(s) = 1 | + (0.719 − 0.694i)2-s + (0.0348 − 0.999i)4-s + (0.939 + 0.342i)5-s + (0.990 − 0.139i)7-s + (−0.669 − 0.743i)8-s + (0.913 − 0.406i)10-s + (0.990 − 0.139i)11-s + (0.241 − 0.970i)13-s + (0.615 − 0.788i)14-s + (−0.997 − 0.0697i)16-s + (0.309 − 0.951i)17-s + (−0.104 + 0.994i)19-s + (0.374 − 0.927i)20-s + (0.615 − 0.788i)22-s + (−0.615 + 0.788i)23-s + ⋯ |
L(s) = 1 | + (0.719 − 0.694i)2-s + (0.0348 − 0.999i)4-s + (0.939 + 0.342i)5-s + (0.990 − 0.139i)7-s + (−0.669 − 0.743i)8-s + (0.913 − 0.406i)10-s + (0.990 − 0.139i)11-s + (0.241 − 0.970i)13-s + (0.615 − 0.788i)14-s + (−0.997 − 0.0697i)16-s + (0.309 − 0.951i)17-s + (−0.104 + 0.994i)19-s + (0.374 − 0.927i)20-s + (0.615 − 0.788i)22-s + (−0.615 + 0.788i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.181 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.181 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.285022963 - 1.901170098i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.285022963 - 1.901170098i\) |
\(L(1)\) |
\(\approx\) |
\(1.773201648 - 0.9030389752i\) |
\(L(1)\) |
\(\approx\) |
\(1.773201648 - 0.9030389752i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 31 | \( 1 \) |
good | 2 | \( 1 + (0.719 - 0.694i)T \) |
| 5 | \( 1 + (0.939 + 0.342i)T \) |
| 7 | \( 1 + (0.990 - 0.139i)T \) |
| 11 | \( 1 + (0.990 - 0.139i)T \) |
| 13 | \( 1 + (0.241 - 0.970i)T \) |
| 17 | \( 1 + (0.309 - 0.951i)T \) |
| 19 | \( 1 + (-0.104 + 0.994i)T \) |
| 23 | \( 1 + (-0.615 + 0.788i)T \) |
| 29 | \( 1 + (-0.719 + 0.694i)T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 + (0.997 - 0.0697i)T \) |
| 43 | \( 1 + (-0.961 - 0.275i)T \) |
| 47 | \( 1 + (-0.559 + 0.829i)T \) |
| 53 | \( 1 + (-0.978 + 0.207i)T \) |
| 59 | \( 1 + (0.719 + 0.694i)T \) |
| 61 | \( 1 + (0.939 - 0.342i)T \) |
| 67 | \( 1 + (0.766 - 0.642i)T \) |
| 71 | \( 1 + (-0.669 - 0.743i)T \) |
| 73 | \( 1 + (-0.309 - 0.951i)T \) |
| 79 | \( 1 + (-0.848 + 0.529i)T \) |
| 83 | \( 1 + (0.961 + 0.275i)T \) |
| 89 | \( 1 + (-0.978 - 0.207i)T \) |
| 97 | \( 1 + (-0.374 + 0.927i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−22.07053011525155807565933374343, −21.72857281864570459533816206943, −20.955024037130852215826690869816, −20.30407717062091466943077472195, −19.05095541749407631242488986196, −17.93578562431491174271422236317, −17.32989649875744933063812448794, −16.8120305511512921659726087277, −15.8913851236541355251643693640, −14.67812617300132610481387772594, −14.43382486137132244201318266895, −13.56015548764309177377406846459, −12.7580700658404170743053028290, −11.82869857615683203485826613065, −11.152746183929560727070231822181, −9.80481076205117893650585871775, −8.800351419647016216746143738464, −8.31731233545074514395401280022, −7.00200673977974805576376363067, −6.32217488141452867828287164193, −5.46587079253833041954512771008, −4.56469664897895003249442505138, −3.87854707937193595731222492094, −2.33439542759018139539168561663, −1.56577468219085777802129094949,
1.24108166310360565429144370153, 1.858267794063679703532532002554, 3.07164345671062108711542714303, 3.89870055069939771873198584499, 5.14180219745982930820129924377, 5.66111468909186965624355290604, 6.621474844723280788474770968607, 7.74086008616764885833328931686, 9.00749228917604606648197500620, 9.82219686043390576847548258225, 10.60422915112875510469525043105, 11.3570048346739555456078976579, 12.141150243530059634197240380846, 13.09673253384015415644865334285, 14.03826957269601286752381190685, 14.3341390074013108601810610326, 15.14626231521452149280961043617, 16.29988115631827662338351010721, 17.42297283629243836183418540491, 18.04623992665625246890343396130, 18.773874049402266600749165859943, 19.8139928231664638767215140783, 20.685177010537981365283162146944, 21.019781096954050855411277640905, 22.05525087369424878115055577299