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
−22.05525087369424878115055577299, −21.019781096954050855411277640905, −20.685177010537981365283162146944, −19.8139928231664638767215140783, −18.773874049402266600749165859943, −18.04623992665625246890343396130, −17.42297283629243836183418540491, −16.29988115631827662338351010721, −15.14626231521452149280961043617, −14.3341390074013108601810610326, −14.03826957269601286752381190685, −13.09673253384015415644865334285, −12.141150243530059634197240380846, −11.3570048346739555456078976579, −10.60422915112875510469525043105, −9.82219686043390576847548258225, −9.00749228917604606648197500620, −7.74086008616764885833328931686, −6.621474844723280788474770968607, −5.66111468909186965624355290604, −5.14180219745982930820129924377, −3.89870055069939771873198584499, −3.07164345671062108711542714303, −1.858267794063679703532532002554, −1.24108166310360565429144370153,
1.56577468219085777802129094949, 2.33439542759018139539168561663, 3.87854707937193595731222492094, 4.56469664897895003249442505138, 5.46587079253833041954512771008, 6.32217488141452867828287164193, 7.00200673977974805576376363067, 8.31731233545074514395401280022, 8.800351419647016216746143738464, 9.80481076205117893650585871775, 11.152746183929560727070231822181, 11.82869857615683203485826613065, 12.7580700658404170743053028290, 13.56015548764309177377406846459, 14.43382486137132244201318266895, 14.67812617300132610481387772594, 15.8913851236541355251643693640, 16.8120305511512921659726087277, 17.32989649875744933063812448794, 17.93578562431491174271422236317, 19.05095541749407631242488986196, 20.30407717062091466943077472195, 20.955024037130852215826690869816, 21.72857281864570459533816206943, 22.07053011525155807565933374343