| L(s) = 1 | + (−85.3 − 147. i)3-s + (−1.41e3 − 817. i)5-s + (796. − 6.30e3i)7-s + (−4.72e3 + 8.19e3i)9-s + (−7.60e4 + 4.38e4i)11-s − 2.48e4i·13-s + 2.79e5i·15-s + (−9.87e4 + 5.70e4i)17-s + (−1.20e5 + 2.08e5i)19-s + (−9.99e5 + 4.20e5i)21-s + (−1.37e6 − 7.95e5i)23-s + (3.59e5 + 6.23e5i)25-s − 1.74e6·27-s − 1.17e5·29-s + (−1.04e6 − 1.80e6i)31-s + ⋯ |
| L(s) = 1 | + (−0.608 − 1.05i)3-s + (−1.01 − 0.584i)5-s + (0.125 − 0.992i)7-s + (−0.240 + 0.416i)9-s + (−1.56 + 0.903i)11-s − 0.241i·13-s + 1.42i·15-s + (−0.286 + 0.165i)17-s + (−0.211 + 0.366i)19-s + (−1.12 + 0.471i)21-s + (−1.02 − 0.593i)23-s + (0.184 + 0.319i)25-s − 0.632·27-s − 0.0307·29-s + (−0.202 − 0.351i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.963 - 0.267i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.963 - 0.267i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.1110431856\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1110431856\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (-796. + 6.30e3i)T \) |
| good | 3 | \( 1 + (85.3 + 147. i)T + (-9.84e3 + 1.70e4i)T^{2} \) |
| 5 | \( 1 + (1.41e3 + 817. i)T + (9.76e5 + 1.69e6i)T^{2} \) |
| 11 | \( 1 + (7.60e4 - 4.38e4i)T + (1.17e9 - 2.04e9i)T^{2} \) |
| 13 | \( 1 + 2.48e4iT - 1.06e10T^{2} \) |
| 17 | \( 1 + (9.87e4 - 5.70e4i)T + (5.92e10 - 1.02e11i)T^{2} \) |
| 19 | \( 1 + (1.20e5 - 2.08e5i)T + (-1.61e11 - 2.79e11i)T^{2} \) |
| 23 | \( 1 + (1.37e6 + 7.95e5i)T + (9.00e11 + 1.55e12i)T^{2} \) |
| 29 | \( 1 + 1.17e5T + 1.45e13T^{2} \) |
| 31 | \( 1 + (1.04e6 + 1.80e6i)T + (-1.32e13 + 2.28e13i)T^{2} \) |
| 37 | \( 1 + (-1.68e6 + 2.92e6i)T + (-6.49e13 - 1.12e14i)T^{2} \) |
| 41 | \( 1 - 2.72e7iT - 3.27e14T^{2} \) |
| 43 | \( 1 + 1.43e7iT - 5.02e14T^{2} \) |
| 47 | \( 1 + (-1.34e7 + 2.33e7i)T + (-5.59e14 - 9.69e14i)T^{2} \) |
| 53 | \( 1 + (3.45e7 + 5.98e7i)T + (-1.64e15 + 2.85e15i)T^{2} \) |
| 59 | \( 1 + (1.41e7 + 2.45e7i)T + (-4.33e15 + 7.50e15i)T^{2} \) |
| 61 | \( 1 + (-1.15e8 - 6.67e7i)T + (5.84e15 + 1.01e16i)T^{2} \) |
| 67 | \( 1 + (-4.49e7 + 2.59e7i)T + (1.36e16 - 2.35e16i)T^{2} \) |
| 71 | \( 1 + 3.87e8iT - 4.58e16T^{2} \) |
| 73 | \( 1 + (-4.09e8 + 2.36e8i)T + (2.94e16 - 5.09e16i)T^{2} \) |
| 79 | \( 1 + (-3.35e8 - 1.93e8i)T + (5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 + 5.59e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + (-4.49e8 - 2.59e8i)T + (1.75e17 + 3.03e17i)T^{2} \) |
| 97 | \( 1 - 2.35e8iT - 7.60e17T^{2} \) |
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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
−12.12262620803532902810836639683, −11.00935309628073112820224621132, −10.00109565300475207485588112316, −8.047052777752460866280959085607, −7.68879061248977189212107132753, −6.54977599922790311280063405498, −5.05864992001070082230949659895, −3.95940175283820897101500732449, −2.01656273418377076036298464296, −0.59854985659997796229616632516,
0.05348303055138803803432057702, 2.53526533753772461479533263231, 3.74452891032056114988625339312, 5.00702203473086959984855805319, 5.89462232587827055652719710808, 7.55282003081312466368228535919, 8.569135546798604272917779657748, 9.895744646820529628024409931538, 11.01493935877106208485649395607, 11.35436946533676121217915459673
