| L(s) = 1 | − 5.90i·2-s − 2.89·4-s + 129.·7-s − 171. i·8-s − 511. i·11-s − 765. i·14-s − 1.10e3·16-s − 3.02e3·22-s − 2.71e3i·23-s − 3.12e3·25-s − 375.·28-s + 1.34e3i·29-s + 1.04e3i·32-s + 1.40e4·37-s + 2.12e4·43-s + 1.48e3i·44-s + ⋯ |
| L(s) = 1 | − 1.04i·2-s − 0.0905·4-s + 0.999·7-s − 0.949i·8-s − 1.27i·11-s − 1.04i·14-s − 1.08·16-s − 1.33·22-s − 1.07i·23-s − 25-s − 0.0905·28-s + 0.295i·29-s + 0.180i·32-s + 1.69·37-s + 1.74·43-s + 0.115i·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.900370 - 1.73938i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.900370 - 1.73938i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - 129.T \) |
| good | 2 | \( 1 + 5.90iT - 32T^{2} \) |
| 5 | \( 1 + 3.12e3T^{2} \) |
| 11 | \( 1 + 511. iT - 1.61e5T^{2} \) |
| 13 | \( 1 - 3.71e5T^{2} \) |
| 17 | \( 1 + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.47e6T^{2} \) |
| 23 | \( 1 + 2.71e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 1.34e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 2.86e7T^{2} \) |
| 37 | \( 1 - 1.40e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.12e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.29e8T^{2} \) |
| 53 | \( 1 - 4.04e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 7.14e8T^{2} \) |
| 61 | \( 1 - 8.44e8T^{2} \) |
| 67 | \( 1 + 6.93e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.16e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 - 2.07e9T^{2} \) |
| 79 | \( 1 - 8.01e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.93e9T^{2} \) |
| 89 | \( 1 + 5.58e9T^{2} \) |
| 97 | \( 1 - 8.58e9T^{2} \) |
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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
−13.41623049249028222065329076146, −12.17547585665243045230008246280, −11.24323235916671759247936945772, −10.55924888300124666844664091195, −9.069317147761003543689592595978, −7.74300661808607710126112836458, −6.01900396093620584601112936172, −4.19062686112063448894023097600, −2.58756870774823721231766080818, −0.973104421258164005292217519145,
1.99651099752960050342563149254, 4.54770436290797150410045741235, 5.80949115446361750107775958106, 7.29047198770844664996350294741, 8.011580938823639771925884609577, 9.520667836780700686675666933757, 11.06613803983129030153462960461, 12.04522285799127052084115219290, 13.59631299856864238295087620849, 14.73215570575066711309450768748
