| L(s) = 1 | − 9i·3-s + 42.1i·5-s + 139.·7-s − 81·9-s + 42.1i·11-s − 424. i·13-s + 379.·15-s − 100.·17-s + 879. i·19-s − 1.25e3i·21-s + 2.75e3·23-s + 1.34e3·25-s + 729i·27-s − 5.48e3i·29-s − 5.48e3·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 0.754i·5-s + 1.07·7-s − 0.333·9-s + 0.105i·11-s − 0.695i·13-s + 0.435·15-s − 0.0847·17-s + 0.558i·19-s − 0.621i·21-s + 1.08·23-s + 0.430·25-s + 0.192i·27-s − 1.21i·29-s − 1.02·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.427724910\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.427724910\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 9iT \) |
| good | 5 | \( 1 - 42.1iT - 3.12e3T^{2} \) |
| 7 | \( 1 - 139.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 42.1iT - 1.61e5T^{2} \) |
| 13 | \( 1 + 424. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 100.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 879. iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 2.75e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 5.48e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 5.48e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.03e4iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 9.17e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 6.66e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 2.54e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.49e3iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 2.43e3iT - 7.14e8T^{2} \) |
| 61 | \( 1 - 3.08e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 - 9.60e3iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 7.65e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 7.83e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.93e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.83e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 1.17e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 9.22e4T + 8.58e9T^{2} \) |
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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
−10.79931513928217675046269142389, −9.658689920208445330659069712667, −8.406357712994554813001043471271, −7.72449817382442716564193817839, −6.84919429639200304640353512050, −5.78316755309018842314364242394, −4.71330767055379152951961735895, −3.25777360646729844245885341406, −2.13789885978684166604100076769, −0.910984138239736715890876969824,
0.811263759657787389278640853452, 2.07184122006874892776214492451, 3.65747085288705921388645248457, 4.82480468453204450294622285876, 5.25120953103223105686987150244, 6.77191769799577742735168174931, 7.88120800858284557549553489790, 8.945408155878338787115527055857, 9.261496188452348924473746313900, 10.82669391846517193121547541106
