| L(s) = 1 | − 4i·2-s + 11i·3-s − 16·4-s + 44·6-s − 142i·7-s + 64i·8-s + 122·9-s + 777·11-s − 176i·12-s − 884i·13-s − 568·14-s + 256·16-s − 27i·17-s − 488i·18-s − 1.14e3·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.705i·3-s − 0.5·4-s + 0.498·6-s − 1.09i·7-s + 0.353i·8-s + 0.502·9-s + 1.93·11-s − 0.352i·12-s − 1.45i·13-s − 0.774·14-s + 0.250·16-s − 0.0226i·17-s − 0.355i·18-s − 0.727·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.44526 - 0.893224i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.44526 - 0.893224i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4iT \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 11iT - 243T^{2} \) |
| 7 | \( 1 + 142iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 777T + 1.61e5T^{2} \) |
| 13 | \( 1 + 884iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 27iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 1.14e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.85e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 4.92e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.80e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.31e4iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 1.51e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 7.84e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 6.73e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 3.41e3iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 3.39e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.74e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.31e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 7.54e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.98e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 7.58e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.62e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 3.05e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.04e5iT - 8.58e9T^{2} \) |
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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
−14.29664492482540769971032363903, −13.13418544431965064937595663133, −11.91016735075260244461662309439, −10.54367092631272152638654048337, −9.960881691380765587499545181102, −8.521141816486098992104524450252, −6.72838929914070138888969223904, −4.58946927404198585860441083335, −3.56708311957626427378338067777, −1.04613955397325644467732009188,
1.65269535269292985872141067578, 4.25903662476534630250045728716, 6.20564456855119088105813801043, 6.97533587455500522857941162521, 8.640589394741607108064484111378, 9.498313974832637497772900087250, 11.67225657834063413573916698039, 12.41679335042971930139323340862, 13.81679845914373176574972871616, 14.68774911181443542809661485494
