L(s) = 1 | + 9.64i·2-s − 61.1·4-s − 129.·7-s − 280. i·8-s − 618. i·11-s − 1.25e3i·14-s + 754.·16-s + 5.97e3·22-s + 4.28e3i·23-s − 3.12e3·25-s + 7.92e3·28-s − 8.95e3i·29-s − 1.70e3i·32-s − 1.40e4·37-s − 2.12e4·43-s + 3.78e4i·44-s + ⋯ |
L(s) = 1 | + 1.70i·2-s − 1.90·4-s − 0.999·7-s − 1.55i·8-s − 1.54i·11-s − 1.70i·14-s + 0.736·16-s + 2.62·22-s + 1.68i·23-s − 25-s + 1.90·28-s − 1.97i·29-s − 0.294i·32-s − 1.69·37-s − 1.74·43-s + 2.94i·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
\(L(3)\) |
\(\approx\) |
\(0.120092 - 0.0621647i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.120092 - 0.0621647i\) |
\(L(\frac{7}{2})\) |
|
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 | \( 1 - 9.64iT - 32T^{2} \) |
| 5 | \( 1 + 3.12e3T^{2} \) |
| 11 | \( 1 + 618. iT - 1.61e5T^{2} \) |
| 13 | \( 1 - 3.71e5T^{2} \) |
| 17 | \( 1 + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.47e6T^{2} \) |
| 23 | \( 1 - 4.28e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 8.95e3iT - 2.05e7T^{2} \) |
| 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 | \( 1 - 5.77e3iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 7.14e8T^{2} \) |
| 61 | \( 1 - 8.44e8T^{2} \) |
| 67 | \( 1 + 6.93e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.84e4iT - 1.80e9T^{2} \) |
| 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
−13.70609214899289389771780070565, −13.48644610909755445562613500895, −11.64934873555788605529851249261, −9.867510231274606144663985709756, −8.760011926954353908878191801487, −7.65114747993662282306935125514, −6.35166431519971778809735016045, −5.55148026093950638840237854976, −3.63104878253809233795391251025, −0.06125980103537006769415445664,
1.90127750666614324096065757952, 3.34697196855803505590089160490, 4.74319531019703766701596170577, 6.87187161668039822808593681308, 8.856189770669661414741722677776, 9.919788861108334987207193432950, 10.58541898733754119722701111743, 12.14488914185738848744225463998, 12.56748046271307411056521251843, 13.64541111539118848865764685108