| L(s) = 1 | + (−26.9 + 15.5i)2-s + (20.0 + 34.7i)3-s + (228. − 394. i)4-s − 2.23e3i·5-s + (−1.08e3 − 623. i)6-s + (−2.07e3 − 1.20e3i)7-s − 1.74e3i·8-s + (9.03e3 − 1.56e4i)9-s + (3.46e4 + 6.00e4i)10-s + (2.22e4 − 1.28e4i)11-s + 1.82e4·12-s + (2.99e4 − 9.85e4i)13-s + 7.47e4·14-s + (7.74e4 − 4.47e4i)15-s + (1.43e5 + 2.49e5i)16-s + (3.43e5 − 5.95e5i)17-s + ⋯ |
| L(s) = 1 | + (−1.19 + 0.687i)2-s + (0.142 + 0.247i)3-s + (0.445 − 0.771i)4-s − 1.59i·5-s + (−0.340 − 0.196i)6-s + (−0.327 − 0.188i)7-s − 0.150i·8-s + (0.459 − 0.795i)9-s + (1.09 + 1.90i)10-s + (0.458 − 0.264i)11-s + 0.254·12-s + (0.291 − 0.956i)13-s + 0.519·14-s + (0.395 − 0.228i)15-s + (0.548 + 0.950i)16-s + (0.998 − 1.72i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.303 + 0.952i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.303 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.656209 - 0.897488i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.656209 - 0.897488i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (2.07e3 + 1.20e3i)T \) |
| 13 | \( 1 + (-2.99e4 + 9.85e4i)T \) |
| good | 2 | \( 1 + (26.9 - 15.5i)T + (256 - 443. i)T^{2} \) |
| 3 | \( 1 + (-20.0 - 34.7i)T + (-9.84e3 + 1.70e4i)T^{2} \) |
| 5 | \( 1 + 2.23e3iT - 1.95e6T^{2} \) |
| 11 | \( 1 + (-2.22e4 + 1.28e4i)T + (1.17e9 - 2.04e9i)T^{2} \) |
| 17 | \( 1 + (-3.43e5 + 5.95e5i)T + (-5.92e10 - 1.02e11i)T^{2} \) |
| 19 | \( 1 + (-1.81e5 - 1.04e5i)T + (1.61e11 + 2.79e11i)T^{2} \) |
| 23 | \( 1 + (1.04e6 + 1.81e6i)T + (-9.00e11 + 1.55e12i)T^{2} \) |
| 29 | \( 1 + (-3.04e6 - 5.27e6i)T + (-7.25e12 + 1.25e13i)T^{2} \) |
| 31 | \( 1 + 8.38e5iT - 2.64e13T^{2} \) |
| 37 | \( 1 + (4.53e6 - 2.62e6i)T + (6.49e13 - 1.12e14i)T^{2} \) |
| 41 | \( 1 + (-2.06e7 + 1.19e7i)T + (1.63e14 - 2.83e14i)T^{2} \) |
| 43 | \( 1 + (1.45e7 - 2.51e7i)T + (-2.51e14 - 4.35e14i)T^{2} \) |
| 47 | \( 1 + 3.03e7iT - 1.11e15T^{2} \) |
| 53 | \( 1 - 3.67e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + (7.05e7 + 4.07e7i)T + (4.33e15 + 7.50e15i)T^{2} \) |
| 61 | \( 1 + (-1.33e7 + 2.30e7i)T + (-5.84e15 - 1.01e16i)T^{2} \) |
| 67 | \( 1 + (-1.94e8 + 1.12e8i)T + (1.36e16 - 2.35e16i)T^{2} \) |
| 71 | \( 1 + (-1.22e8 - 7.04e7i)T + (2.29e16 + 3.97e16i)T^{2} \) |
| 73 | \( 1 + 2.61e8iT - 5.88e16T^{2} \) |
| 79 | \( 1 - 4.67e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 2.60e8iT - 1.86e17T^{2} \) |
| 89 | \( 1 + (-1.95e8 + 1.12e8i)T + (1.75e17 - 3.03e17i)T^{2} \) |
| 97 | \( 1 + (1.62e8 + 9.38e7i)T + (3.80e17 + 6.58e17i)T^{2} \) |
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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
−12.13602315878098811723545866163, −10.22777731325657921897712550566, −9.418770853589111841975427288169, −8.722645598790908339975334866457, −7.75387880333105518590449233457, −6.44610326896762245478139257389, −5.02916953731457009023800814978, −3.55784606518335964803459540862, −0.960281323829791782220869975956, −0.61012689983958388188049963922,
1.48504417352441584405567672226, 2.37490835222675615680763021690, 3.73207098497212777884918790733, 6.04677734020958138658737669394, 7.27254369736761062646075491776, 8.174330686103199364356436663747, 9.655069462544239082852799459353, 10.29379292073651592572973125310, 11.18862072081604568198518230317, 12.12144817346759834727620765504
