L(s) = 1 | + (−0.535 + 0.844i)2-s + (0.728 − 0.684i)3-s + (−0.425 − 0.904i)4-s + (0.187 + 0.982i)6-s + (0.809 − 0.587i)7-s + (0.992 + 0.125i)8-s + (0.0627 − 0.998i)9-s + (−0.929 − 0.368i)12-s + (−0.0627 + 0.998i)13-s + (0.0627 + 0.998i)14-s + (−0.637 + 0.770i)16-s + (0.425 − 0.904i)17-s + (0.809 + 0.587i)18-s + (−0.728 − 0.684i)19-s + (0.187 − 0.982i)21-s + ⋯ |
L(s) = 1 | + (−0.535 + 0.844i)2-s + (0.728 − 0.684i)3-s + (−0.425 − 0.904i)4-s + (0.187 + 0.982i)6-s + (0.809 − 0.587i)7-s + (0.992 + 0.125i)8-s + (0.0627 − 0.998i)9-s + (−0.929 − 0.368i)12-s + (−0.0627 + 0.998i)13-s + (0.0627 + 0.998i)14-s + (−0.637 + 0.770i)16-s + (0.425 − 0.904i)17-s + (0.809 + 0.587i)18-s + (−0.728 − 0.684i)19-s + (0.187 − 0.982i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0251i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0251i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.003010496497 - 0.2395550889i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.003010496497 - 0.2395550889i\) |
\(L(1)\) |
\(\approx\) |
\(0.9185458994 + 0.01605978112i\) |
\(L(1)\) |
\(\approx\) |
\(0.9185458994 + 0.01605978112i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.535 + 0.844i)T \) |
| 3 | \( 1 + (0.728 - 0.684i)T \) |
| 7 | \( 1 + (0.809 - 0.587i)T \) |
| 13 | \( 1 + (-0.0627 + 0.998i)T \) |
| 17 | \( 1 + (0.425 - 0.904i)T \) |
| 19 | \( 1 + (-0.728 - 0.684i)T \) |
| 23 | \( 1 + (0.968 + 0.248i)T \) |
| 29 | \( 1 + (-0.876 - 0.481i)T \) |
| 31 | \( 1 + (-0.425 + 0.904i)T \) |
| 37 | \( 1 + (-0.637 + 0.770i)T \) |
| 41 | \( 1 + (-0.968 + 0.248i)T \) |
| 43 | \( 1 + (-0.309 + 0.951i)T \) |
| 47 | \( 1 + (-0.992 + 0.125i)T \) |
| 53 | \( 1 + (-0.187 + 0.982i)T \) |
| 59 | \( 1 + (-0.929 - 0.368i)T \) |
| 61 | \( 1 + (-0.968 - 0.248i)T \) |
| 67 | \( 1 + (0.876 - 0.481i)T \) |
| 71 | \( 1 + (-0.992 + 0.125i)T \) |
| 73 | \( 1 + (0.929 - 0.368i)T \) |
| 79 | \( 1 + (-0.728 + 0.684i)T \) |
| 83 | \( 1 + (-0.728 - 0.684i)T \) |
| 89 | \( 1 + (-0.929 + 0.368i)T \) |
| 97 | \( 1 + (0.876 + 0.481i)T \) |
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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
−20.964088795241492159542544554381, −20.389539684203814167415063075731, −19.60247941090349774784508036550, −18.82859746946462871899510253660, −18.30193657529574273547599541016, −17.15557137867779267519398880171, −16.80369491105466163482224018886, −15.59328912609208898690171986773, −14.89979740538255753898683682589, −14.310759891226924889957465933044, −13.12718618187902465161274108528, −12.64844773956305301988011744645, −11.60284963580772928221689471956, −10.71859178567863295511062357448, −10.33746772850025138243838573688, −9.34156528578612115349399355997, −8.57408877008094056492450394103, −8.13523025680255291483070766311, −7.31304105771971449315903620133, −5.64738101938406719357832291560, −4.88580877280086239303574681121, −3.862850665813661627199796507508, −3.216964176470503115784457277985, −2.17882294063017014861000029096, −1.54031351470501659643410079480,
0.04700904304262043029316088945, 1.24712436097136330516783424634, 1.837632014294365340528803622971, 3.16801668150682963967249265426, 4.41918405667156984766953067743, 5.083189266255000705964662934935, 6.39717491848738661621301087182, 7.032718912883328755099905054, 7.60225029076972308628285423102, 8.44471422921467147362821514886, 9.11069328011492454139757285757, 9.82261558113991249747399297565, 10.97423283621735569621097744497, 11.65648403076064101430440784031, 12.902020188712625284074633681681, 13.71600336235831266783744162936, 14.17445348734109799450585344163, 14.90502054604269111450206091455, 15.56463985852751154710739291895, 16.76734935472218168574170952991, 17.14744382450682896569943986412, 18.11568431397361892573634258673, 18.6244470905165864287694016909, 19.368649722953340209633374086375, 20.05424723934485969132088925195