| L(s) = 1 | − 2.26·3-s − i·5-s + 1.11i·7-s + 2.11·9-s + 5.37i·11-s + (−3.37 + 1.26i)13-s + 2.26i·15-s + 7.90i·19-s − 2.52i·21-s − 6.49·23-s − 25-s + 2·27-s + 3.63·29-s − 3.14i·31-s − 12.1i·33-s + ⋯ |
| L(s) = 1 | − 1.30·3-s − 0.447i·5-s + 0.421i·7-s + 0.705·9-s + 1.62i·11-s + (−0.936 + 0.349i)13-s + 0.583i·15-s + 1.81i·19-s − 0.550i·21-s − 1.35·23-s − 0.200·25-s + 0.384·27-s + 0.675·29-s − 0.565i·31-s − 2.11i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.349 - 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.349 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.282042 + 0.406448i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.282042 + 0.406448i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + iT \) |
| 13 | \( 1 + (3.37 - 1.26i)T \) |
| good | 3 | \( 1 + 2.26T + 3T^{2} \) |
| 7 | \( 1 - 1.11iT - 7T^{2} \) |
| 11 | \( 1 - 5.37iT - 11T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 7.90iT - 19T^{2} \) |
| 23 | \( 1 + 6.49T + 23T^{2} \) |
| 29 | \( 1 - 3.63T + 29T^{2} \) |
| 31 | \( 1 + 3.14iT - 31T^{2} \) |
| 37 | \( 1 - 7.40iT - 37T^{2} \) |
| 41 | \( 1 + 4.75iT - 41T^{2} \) |
| 43 | \( 1 + 4.78T + 43T^{2} \) |
| 47 | \( 1 + 6.16iT - 47T^{2} \) |
| 53 | \( 1 - 0.292T + 53T^{2} \) |
| 59 | \( 1 + 11.3iT - 59T^{2} \) |
| 61 | \( 1 + 5.34T + 61T^{2} \) |
| 67 | \( 1 - 11.8iT - 67T^{2} \) |
| 71 | \( 1 - 3.43iT - 71T^{2} \) |
| 73 | \( 1 - 4.59iT - 73T^{2} \) |
| 79 | \( 1 + 10.8T + 79T^{2} \) |
| 83 | \( 1 + 7.40iT - 83T^{2} \) |
| 89 | \( 1 - 14.8iT - 89T^{2} \) |
| 97 | \( 1 + 3.76iT - 97T^{2} \) |
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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
−12.10977350944044636678888220779, −11.74011611367768377829511017224, −10.15271561514571770410132780729, −9.878601338544687943855587556675, −8.336880769128702584993946747788, −7.19554952186384732275227899495, −6.10694181351273231024228946637, −5.17422814019535993076890703684, −4.25599268122250891520053805983, −1.93901757859312782630584221894,
0.44046045040546209234330452445, 2.95807887502433541624028549499, 4.58746939953694681042598312506, 5.69118701232763809820752689604, 6.49953412513349340777205932718, 7.54311122789817694711036219218, 8.833709285733065172238730399308, 10.17450068876300009305252540840, 10.89749419927772390565708541732, 11.51467276811020185216378042810
