| 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
−11.35436946533676121217915459673, −11.01493935877106208485649395607, −9.895744646820529628024409931538, −8.569135546798604272917779657748, −7.55282003081312466368228535919, −5.89462232587827055652719710808, −5.00702203473086959984855805319, −3.74452891032056114988625339312, −2.53526533753772461479533263231, −0.05348303055138803803432057702,
0.59854985659997796229616632516, 2.01656273418377076036298464296, 3.95940175283820897101500732449, 5.05864992001070082230949659895, 6.54977599922790311280063405498, 7.68879061248977189212107132753, 8.047052777752460866280959085607, 10.00109565300475207485588112316, 11.00935309628073112820224621132, 12.12262620803532902810836639683
