| L(s) = 1 | + 15.2i·3-s + 43.0i·5-s − 22.6·7-s + 10.9·9-s + 258. i·11-s + 990. i·13-s − 656.·15-s − 218·17-s − 1.99e3i·19-s − 344. i·21-s − 3.41e3·23-s + 1.26e3·25-s + 3.86e3i·27-s − 4.86e3i·29-s − 8.68e3·31-s + ⋯ |
| L(s) = 1 | + 0.977i·3-s + 0.770i·5-s − 0.174·7-s + 0.0452·9-s + 0.645i·11-s + 1.62i·13-s − 0.753·15-s − 0.182·17-s − 1.26i·19-s − 0.170i·21-s − 1.34·23-s + 0.406·25-s + 1.02i·27-s − 1.07i·29-s − 1.62·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.235695224\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.235695224\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 15.2iT - 243T^{2} \) |
| 5 | \( 1 - 43.0iT - 3.12e3T^{2} \) |
| 7 | \( 1 + 22.6T + 1.68e4T^{2} \) |
| 11 | \( 1 - 258. iT - 1.61e5T^{2} \) |
| 13 | \( 1 - 990. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 218T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.99e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 + 3.41e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.86e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 8.68e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 3.23e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 8.41e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 685. iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 1.81e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.65e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 3.37e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 - 2.73e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 - 6.67e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 2.16e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.30e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.47e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 7.06e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 9.38e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 8.17e4T + 8.58e9T^{2} \) |
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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
−12.95568532810522549933630562002, −11.62090847913113263804253439143, −10.81741760450958977197232183229, −9.754587050449619558649528923755, −9.113571161627410402262884913747, −7.37040938110388136311826466785, −6.43288505140980835102427535840, −4.73560640896430146023574519629, −3.81641555023300577896690658918, −2.16935530469218808983923464743,
0.43313630730099954934755159652, 1.67916974674188387049974782690, 3.49379594111043727230773498518, 5.26800134364843032966843893458, 6.32231315479664655308840849008, 7.74121945946907517993164589379, 8.370701166970897100980089055589, 9.787970093015442501428400681667, 10.92193573230787133953391966566, 12.44341441994242617458260362906
