| L(s) = 1 | + (10.5 + 11.4i)3-s − 101. i·5-s − 13.9i·7-s + (−18.9 + 242. i)9-s − 445.·11-s − 217.·13-s + (1.16e3 − 1.07e3i)15-s − 1.43e3i·17-s + 2.61e3i·19-s + (159. − 147. i)21-s + 696.·23-s − 7.26e3·25-s + (−2.97e3 + 2.34e3i)27-s − 4.60e3i·29-s + 8.33e3i·31-s + ⋯ |
| L(s) = 1 | + (0.679 + 0.734i)3-s − 1.82i·5-s − 0.107i·7-s + (−0.0778 + 0.996i)9-s − 1.11·11-s − 0.356·13-s + (1.33 − 1.23i)15-s − 1.20i·17-s + 1.65i·19-s + (0.0789 − 0.0730i)21-s + 0.274·23-s − 2.32·25-s + (−0.784 + 0.619i)27-s − 1.01i·29-s + 1.55i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.679 - 0.734i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.679 - 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.6993780087\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6993780087\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-10.5 - 11.4i)T \) |
| good | 5 | \( 1 + 101. iT - 3.12e3T^{2} \) |
| 7 | \( 1 + 13.9iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 445.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 217.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.43e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 2.61e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 696.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.60e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 8.33e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 + 6.92e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.16e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 - 6.05e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 1.30e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.78e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 4.11e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 203.T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.28e3iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 4.54e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.24e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.85e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 + 7.61e3T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.52e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 + 8.78e4T + 8.58e9T^{2} \) |
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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
−10.57130627788068787929508144551, −9.780499067427624223947689822950, −9.036818326766661772603161020478, −8.213009182832269477464240669828, −7.57517496915959406703454729897, −5.58000832881859303527102671073, −4.95389185323240183096360603557, −4.10173010534511293170195487841, −2.71249866725493465918615648399, −1.32282680832229242800516338696,
0.14982414725161840758410244588, 2.14418100515932016058676264106, 2.75245682919298582416923197629, 3.76693749437072889683836682192, 5.57925399855751126752218851357, 6.73053834190873179258451682966, 7.22981935740111665661067384832, 8.113121491586597772894337724321, 9.219013854610538063478209735559, 10.37924126609017612312033404998
