| L(s) = 1 | + 3i·2-s + 7·4-s + 19·7-s + 69i·8-s + 123i·11-s − 302·13-s + 57i·14-s − 95·16-s − 414i·17-s − 304·19-s − 369·22-s − 300i·23-s − 906i·26-s + 133·28-s − 678i·29-s + ⋯ |
| L(s) = 1 | + 0.750i·2-s + 0.437·4-s + 0.387·7-s + 1.07i·8-s + 1.01i·11-s − 1.78·13-s + 0.290i·14-s − 0.371·16-s − 1.43i·17-s − 0.842·19-s − 0.762·22-s − 0.567i·23-s − 1.34i·26-s + 0.169·28-s − 0.806i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.4163601347\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4163601347\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 3iT - 16T^{2} \) |
| 7 | \( 1 - 19T + 2.40e3T^{2} \) |
| 11 | \( 1 - 123iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 302T + 2.85e4T^{2} \) |
| 17 | \( 1 + 414iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 304T + 1.30e5T^{2} \) |
| 23 | \( 1 + 300iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 678iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 239T + 9.23e5T^{2} \) |
| 37 | \( 1 + 740T + 1.87e6T^{2} \) |
| 41 | \( 1 - 228iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 982T + 3.41e6T^{2} \) |
| 47 | \( 1 + 2.16e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 1.59e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 2.92e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 316T + 1.38e7T^{2} \) |
| 67 | \( 1 + 4.62e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 1.81e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 3.03e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 1.04e4T + 3.89e7T^{2} \) |
| 83 | \( 1 + 1.26e4iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 7.00e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 6.51e3T + 8.85e7T^{2} \) |
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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
−9.711485517390404764605987718302, −8.669605622272984441902298112740, −7.55916340705810786071078422778, −7.24272859380829228357702367652, −6.29766966208342011928887389922, −5.06360277203457127580262004644, −4.58784648441129389211185883849, −2.69874635836611166621195919719, −1.99794431432741087962938408326, −0.087575257378152819027457939941,
1.35428194835009892613267935815, 2.34683780288721629468747773724, 3.34722614431845558024134152068, 4.42229016095445448300655023326, 5.63683331087479469110952731380, 6.59710864178066065721455071423, 7.53716008476026591427285031340, 8.429705711514847704548628495144, 9.476326428293756090771939041415, 10.39038210330829854703644816778
