L(s) = 1 | + (−0.648 + 1.60i)3-s − i·5-s + 3.10i·7-s + (−2.15 − 2.08i)9-s + 3.67·11-s + 13-s + (1.60 + 0.648i)15-s + 3.22i·17-s − 7.68i·19-s + (−4.99 − 2.01i)21-s + 8.17·23-s − 25-s + (4.74 − 2.11i)27-s − 1.87i·29-s − 5.90i·31-s + ⋯ |
L(s) = 1 | + (−0.374 + 0.927i)3-s − 0.447i·5-s + 1.17i·7-s + (−0.719 − 0.694i)9-s + 1.10·11-s + 0.277·13-s + (0.414 + 0.167i)15-s + 0.781i·17-s − 1.76i·19-s + (−1.08 − 0.439i)21-s + 1.70·23-s − 0.200·25-s + (0.913 − 0.407i)27-s − 0.347i·29-s − 1.06i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.787 - 0.615i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.787 - 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.738255508\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.738255508\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.648 - 1.60i)T \) |
| 5 | \( 1 + iT \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 - 3.10iT - 7T^{2} \) |
| 11 | \( 1 - 3.67T + 11T^{2} \) |
| 17 | \( 1 - 3.22iT - 17T^{2} \) |
| 19 | \( 1 + 7.68iT - 19T^{2} \) |
| 23 | \( 1 - 8.17T + 23T^{2} \) |
| 29 | \( 1 + 1.87iT - 29T^{2} \) |
| 31 | \( 1 + 5.90iT - 31T^{2} \) |
| 37 | \( 1 + 2.51T + 37T^{2} \) |
| 41 | \( 1 - 2.35iT - 41T^{2} \) |
| 43 | \( 1 + 4.16iT - 43T^{2} \) |
| 47 | \( 1 - 3.77T + 47T^{2} \) |
| 53 | \( 1 + 10.1iT - 53T^{2} \) |
| 59 | \( 1 - 2.06T + 59T^{2} \) |
| 61 | \( 1 - 5.31T + 61T^{2} \) |
| 67 | \( 1 - 14.9iT - 67T^{2} \) |
| 71 | \( 1 + 9.03T + 71T^{2} \) |
| 73 | \( 1 - 4.78T + 73T^{2} \) |
| 79 | \( 1 + 7.58iT - 79T^{2} \) |
| 83 | \( 1 - 4.73T + 83T^{2} \) |
| 89 | \( 1 + 9.91iT - 89T^{2} \) |
| 97 | \( 1 + 4.27T + 97T^{2} \) |
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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
−8.899143780218893063665497775250, −8.487099367768948096461950629621, −7.09890102692226839316568999111, −6.34141453892807038321895524885, −5.62124474841842220692005129413, −4.94270872721360462902450575705, −4.19382619495684545507038019800, −3.28966440492483075186157023006, −2.27939847949858410835386312012, −0.803419854140755402891678388394,
0.911305815070815394182835905266, 1.61617416160849896981120452879, 3.01927570068721934882196593710, 3.77571525682223535213307981655, 4.79716326901821464229159175645, 5.75291829897394100076272383563, 6.57298906418838559775063801348, 7.08164527240029038753541139124, 7.58474065372423158032289544396, 8.503637472009206508610007088013