| L(s) = 1 | + 8i·2-s − 64·4-s + (−117 + 44i)5-s − 512i·8-s − 729·9-s + (−352 − 936i)10-s + 1.65e3i·13-s + 4.09e3·16-s + 9.77e3i·17-s − 5.83e3i·18-s + (7.48e3 − 2.81e3i)20-s + (1.17e4 − 1.02e4i)25-s − 1.32e4·26-s − 3.18e4·29-s + 3.27e4i·32-s + ⋯ |
| L(s) = 1 | + i·2-s − 4-s + (−0.936 + 0.351i)5-s − i·8-s − 0.999·9-s + (−0.351 − 0.936i)10-s + 0.753i·13-s + 16-s + 1.98i·17-s − 0.999i·18-s + (0.936 − 0.351i)20-s + (0.752 − 0.658i)25-s − 0.753·26-s − 1.30·29-s + i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.935 + 0.351i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.935 + 0.351i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.0958016 - 0.526908i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0958016 - 0.526908i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 8iT \) |
| 5 | \( 1 + (117 - 44i)T \) |
| good | 3 | \( 1 + 729T^{2} \) |
| 7 | \( 1 + 1.17e5T^{2} \) |
| 11 | \( 1 - 1.77e6T^{2} \) |
| 13 | \( 1 - 1.65e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 - 9.77e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 - 4.70e7T^{2} \) |
| 23 | \( 1 + 1.48e8T^{2} \) |
| 29 | \( 1 + 3.18e4T + 5.94e8T^{2} \) |
| 31 | \( 1 - 8.87e8T^{2} \) |
| 37 | \( 1 + 8.47e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 - 8.49e4T + 4.75e9T^{2} \) |
| 43 | \( 1 + 6.32e9T^{2} \) |
| 47 | \( 1 + 1.07e10T^{2} \) |
| 53 | \( 1 - 2.96e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 - 4.21e10T^{2} \) |
| 61 | \( 1 + 2.34e5T + 5.15e10T^{2} \) |
| 67 | \( 1 + 9.04e10T^{2} \) |
| 71 | \( 1 - 1.28e11T^{2} \) |
| 73 | \( 1 - 6.50e5iT - 1.51e11T^{2} \) |
| 79 | \( 1 - 2.43e11T^{2} \) |
| 83 | \( 1 + 3.26e11T^{2} \) |
| 89 | \( 1 - 1.37e6T + 4.96e11T^{2} \) |
| 97 | \( 1 + 1.07e6iT - 8.32e11T^{2} \) |
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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
−17.41723559617896442678016846889, −16.40356207080895231034368450412, −15.09591761582450050167333685427, −14.28557198809953293571831263674, −12.58984944327641074052177113615, −10.95791470246594444668912161362, −8.908335115506101954978497364641, −7.67376229366375732381050568330, −6.04677659334590292656429319984, −3.95243560446106331565838514303,
0.34536965349166450433700440595, 3.12628350211487746483549995409, 5.05351961884906686195943630547, 7.949767478799850753433924681023, 9.324844945340345586630063871331, 11.13675731107761517343218195826, 11.97476393214418913467627506173, 13.38409120183690448888657989000, 14.79672026095810305901821197090, 16.41309310941420722358276692430
