| L(s) = 1 | − 16.5i·3-s − 25i·5-s + 55.4·7-s − 31.1·9-s + 153. i·11-s − 57.3i·13-s − 413.·15-s + 1.78e3·17-s − 1.59e3i·19-s − 918. i·21-s + 4.11e3·23-s − 625·25-s − 3.50e3i·27-s + 5.00e3i·29-s + 1.82e3·31-s + ⋯ |
| L(s) = 1 | − 1.06i·3-s − 0.447i·5-s + 0.427·7-s − 0.128·9-s + 0.382i·11-s − 0.0940i·13-s − 0.475·15-s + 1.49·17-s − 1.01i·19-s − 0.454i·21-s + 1.62·23-s − 0.200·25-s − 0.925i·27-s + 1.10i·29-s + 0.340·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.415390873\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.415390873\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 25iT \) |
| good | 3 | \( 1 + 16.5iT - 243T^{2} \) |
| 7 | \( 1 - 55.4T + 1.68e4T^{2} \) |
| 11 | \( 1 - 153. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + 57.3iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 1.78e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.59e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 4.11e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 5.00e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 1.82e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.80e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 5.45e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 8.25e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 8.40e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 6.74e3iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 17.5iT - 7.14e8T^{2} \) |
| 61 | \( 1 + 1.88e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 - 5.97e3iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 1.68e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.66e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.72e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 5.78e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 9.31e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 9.12e4T + 8.58e9T^{2} \) |
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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
−10.62487968583726283576935623430, −9.472114833595491628047913086818, −8.498233583448730888816381119184, −7.50499860103396133802226585982, −6.91221380587420809982157078981, −5.57251893895977716899034318465, −4.60839972623287452856241467029, −2.98740234571001291945862075802, −1.58678953988749660784689739814, −0.75485689567826871008281629860,
1.21054183147506543412362221038, 2.96432725486218213637933293753, 3.92433573027841879341868176235, 5.01256080253922515631293079490, 5.99066814303731666372802188838, 7.34344599155849025574774800730, 8.278482405706656557946853292311, 9.456949488494174369046702379795, 10.13255056166041027637121707225, 10.91508779743395664320656597856
