| L(s) = 1 | − 4i·2-s − 16·4-s + i·7-s + 64i·8-s + 210·11-s − 667i·13-s + 4·14-s + 256·16-s + 114i·17-s − 581·19-s − 840i·22-s + 4.35e3i·23-s − 2.66e3·26-s − 16i·28-s − 126·29-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s + 0.00771i·7-s + 0.353i·8-s + 0.523·11-s − 1.09i·13-s + 0.00545·14-s + 0.250·16-s + 0.0956i·17-s − 0.369·19-s − 0.370i·22-s + 1.71i·23-s − 0.774·26-s − 0.00385i·28-s − 0.0278·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.698919280\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.698919280\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 210T + 1.61e5T^{2} \) |
| 13 | \( 1 + 667iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 114iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 581T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.35e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 126T + 2.05e7T^{2} \) |
| 31 | \( 1 - 7.58e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 3.74e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 2.85e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.82e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 2.33e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 2.16e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 3.23e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 7.16e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.95e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 4.30e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.89e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 2.76e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.78e3iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 5.02e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.42e5iT - 8.58e9T^{2} \) |
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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
−10.12156408226877943621824248343, −9.234944048644015689102158505201, −8.317841410568751854213406685815, −7.37014038105505025701248700503, −6.08108168359572474667640620253, −5.11789048538798468766472068995, −3.91607512273194828883009033144, −2.97980265133117231093747426273, −1.67154239087002161315317696860, −0.48771241351356224643606169378,
1.00151785256436777964068639802, 2.52555578909517756495154906930, 4.08165169623767063795966481191, 4.76951030254582430914644846178, 6.21310094586503380112764106206, 6.67854043315938577220297820730, 7.83084840906536062655923892963, 8.734398903557686475917910150009, 9.460040897411339611506345389348, 10.47037396689332368583978130731
