| L(s) = 1 | − i·3-s − 2.52·5-s + (2.25 − 1.38i)7-s − 9-s + 0.480i·11-s − 0.157i·13-s + 2.52i·15-s + 5.77·17-s + 6.77·19-s + (−1.38 − 2.25i)21-s + (−4.15 − 2.39i)23-s + 1.39·25-s + i·27-s − 0.536·29-s + 8.94i·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 1.13·5-s + (0.852 − 0.522i)7-s − 0.333·9-s + 0.144i·11-s − 0.0437i·13-s + 0.652i·15-s + 1.40·17-s + 1.55·19-s + (−0.301 − 0.492i)21-s + (−0.866 − 0.498i)23-s + 0.278·25-s + 0.192i·27-s − 0.0996·29-s + 1.60i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0277 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0277 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.451125059\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.451125059\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 7 | \( 1 + (-2.25 + 1.38i)T \) |
| 23 | \( 1 + (4.15 + 2.39i)T \) |
| good | 5 | \( 1 + 2.52T + 5T^{2} \) |
| 11 | \( 1 - 0.480iT - 11T^{2} \) |
| 13 | \( 1 + 0.157iT - 13T^{2} \) |
| 17 | \( 1 - 5.77T + 17T^{2} \) |
| 19 | \( 1 - 6.77T + 19T^{2} \) |
| 29 | \( 1 + 0.536T + 29T^{2} \) |
| 31 | \( 1 - 8.94iT - 31T^{2} \) |
| 37 | \( 1 - 1.22iT - 37T^{2} \) |
| 41 | \( 1 + 11.2iT - 41T^{2} \) |
| 43 | \( 1 + 6.31iT - 43T^{2} \) |
| 47 | \( 1 + 1.75iT - 47T^{2} \) |
| 53 | \( 1 + 1.82iT - 53T^{2} \) |
| 59 | \( 1 + 8.60iT - 59T^{2} \) |
| 61 | \( 1 - 14.1T + 61T^{2} \) |
| 67 | \( 1 - 8.94iT - 67T^{2} \) |
| 71 | \( 1 + 8.38T + 71T^{2} \) |
| 73 | \( 1 + 12.2iT - 73T^{2} \) |
| 79 | \( 1 + 16.5iT - 79T^{2} \) |
| 83 | \( 1 + 0.812T + 83T^{2} \) |
| 89 | \( 1 + 14.7T + 89T^{2} \) |
| 97 | \( 1 + 3.39T + 97T^{2} \) |
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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
−8.663460849996313603608891421931, −8.114525715041051282421680199500, −7.38046381197785267255138215660, −7.08370902642252393966052703235, −5.68004963712584137647092269792, −5.00014217577927062627644434975, −3.90653931574194836939982940035, −3.23096857781547172026824333079, −1.74549981233730912045168414491, −0.63617148215646398222092953993,
1.16678553907104378713709285001, 2.74114129114881109424393152434, 3.66572342284037419754225941892, 4.39449817855715386221975154553, 5.35946848359425156206017250023, 5.92635021326666066037198503746, 7.40796362023351531093256882823, 7.88505870553754865628026335674, 8.393961167113478260589053549148, 9.578666083018895642726016943920
