| L(s) = 1 | + (−2.44 + i)7-s − 4.89·11-s − 4·13-s − 4.89i·17-s + 4i·19-s + 6i·23-s − 5·25-s − 4.89i·29-s + 4.89·31-s + 9.79i·37-s − 4.89i·41-s + 9.79·43-s + 9.79·47-s + (4.99 − 4.89i)49-s − 4.89i·53-s + ⋯ |
| L(s) = 1 | + (−0.925 + 0.377i)7-s − 1.47·11-s − 1.10·13-s − 1.18i·17-s + 0.917i·19-s + 1.25i·23-s − 25-s − 0.909i·29-s + 0.879·31-s + 1.61i·37-s − 0.765i·41-s + 1.49·43-s + 1.42·47-s + (0.714 − 0.699i)49-s − 0.672i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.921 + 0.387i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.921 + 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9379953836\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9379953836\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.44 - i)T \) |
| good | 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 + 4.89T + 11T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 + 4.89iT - 17T^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 + 4.89iT - 29T^{2} \) |
| 31 | \( 1 - 4.89T + 31T^{2} \) |
| 37 | \( 1 - 9.79iT - 37T^{2} \) |
| 41 | \( 1 + 4.89iT - 41T^{2} \) |
| 43 | \( 1 - 9.79T + 43T^{2} \) |
| 47 | \( 1 - 9.79T + 47T^{2} \) |
| 53 | \( 1 + 4.89iT - 53T^{2} \) |
| 59 | \( 1 + 12iT - 59T^{2} \) |
| 61 | \( 1 - 4T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 6iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 2iT - 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 + 4.89iT - 89T^{2} \) |
| 97 | \( 1 + 9.79iT - 97T^{2} \) |
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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
−8.235739126859439723418162357307, −7.64973369714447334216620717071, −7.08266405789927779010212943319, −6.04326522742421167656329890836, −5.47147977609412922826870216111, −4.76674401885485970706314345584, −3.66665612391090840924901204751, −2.77537532680608132361348748426, −2.19883452600500595104737780397, −0.43460623550253756369288951559,
0.62158443838516518898616323332, 2.36297227043972424734483420676, 2.78347872471047428140613843628, 3.98000818738421111912639613311, 4.65127548592697585598071378323, 5.62011601407942399753024965989, 6.20952638893700535518816483382, 7.24036903728949152994384936670, 7.54341716478062967220788771354, 8.511554073594217532220355981219
