| L(s) = 1 | + (−6.27 − 1.68i)2-s + (15.1 + 56.4i)3-s + (−406. − 234. i)4-s + (−873. − 1.09e3i)5-s − 379. i·6-s + (259. − 6.34e3i)7-s + (4.50e3 + 4.50e3i)8-s + (1.40e4 − 8.13e3i)9-s + (3.64e3 + 8.30e3i)10-s + (−4.49e4 + 7.78e4i)11-s + (7.10e3 − 2.65e4i)12-s + (6.04e4 − 6.04e4i)13-s + (−1.22e4 + 3.93e4i)14-s + (4.83e4 − 6.58e4i)15-s + (9.95e4 + 1.72e5i)16-s + (−4.52e5 + 1.21e5i)17-s + ⋯ |
| L(s) = 1 | + (−0.277 − 0.0742i)2-s + (0.107 + 0.402i)3-s + (−0.794 − 0.458i)4-s + (−0.625 − 0.780i)5-s − 0.119i·6-s + (0.0408 − 0.999i)7-s + (0.389 + 0.389i)8-s + (0.715 − 0.413i)9-s + (0.115 + 0.262i)10-s + (−0.925 + 1.60i)11-s + (0.0989 − 0.369i)12-s + (0.587 − 0.587i)13-s + (−0.0855 + 0.273i)14-s + (0.246 − 0.335i)15-s + (0.379 + 0.657i)16-s + (−1.31 + 0.352i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.206 - 0.978i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.206 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.245360 + 0.302484i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.245360 + 0.302484i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (873. + 1.09e3i)T \) |
| 7 | \( 1 + (-259. + 6.34e3i)T \) |
| good | 2 | \( 1 + (6.27 + 1.68i)T + (443. + 256i)T^{2} \) |
| 3 | \( 1 + (-15.1 - 56.4i)T + (-1.70e4 + 9.84e3i)T^{2} \) |
| 11 | \( 1 + (4.49e4 - 7.78e4i)T + (-1.17e9 - 2.04e9i)T^{2} \) |
| 13 | \( 1 + (-6.04e4 + 6.04e4i)T - 1.06e10iT^{2} \) |
| 17 | \( 1 + (4.52e5 - 1.21e5i)T + (1.02e11 - 5.92e10i)T^{2} \) |
| 19 | \( 1 + (2.91e5 + 5.04e5i)T + (-1.61e11 + 2.79e11i)T^{2} \) |
| 23 | \( 1 + (4.05e5 - 1.51e6i)T + (-1.55e12 - 9.00e11i)T^{2} \) |
| 29 | \( 1 - 3.27e6iT - 1.45e13T^{2} \) |
| 31 | \( 1 + (-6.78e6 - 3.91e6i)T + (1.32e13 + 2.28e13i)T^{2} \) |
| 37 | \( 1 + (1.62e6 + 4.36e5i)T + (1.12e14 + 6.49e13i)T^{2} \) |
| 41 | \( 1 + 7.67e6iT - 3.27e14T^{2} \) |
| 43 | \( 1 + (-9.94e6 - 9.94e6i)T + 5.02e14iT^{2} \) |
| 47 | \( 1 + (1.83e6 - 6.86e6i)T + (-9.69e14 - 5.59e14i)T^{2} \) |
| 53 | \( 1 + (-5.93e7 + 1.59e7i)T + (2.85e15 - 1.64e15i)T^{2} \) |
| 59 | \( 1 + (3.15e7 - 5.46e7i)T + (-4.33e15 - 7.50e15i)T^{2} \) |
| 61 | \( 1 + (2.44e7 - 1.41e7i)T + (5.84e15 - 1.01e16i)T^{2} \) |
| 67 | \( 1 + (1.16e7 + 4.36e7i)T + (-2.35e16 + 1.36e16i)T^{2} \) |
| 71 | \( 1 + 2.78e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + (-5.00e7 - 1.86e8i)T + (-5.09e16 + 2.94e16i)T^{2} \) |
| 79 | \( 1 + (1.28e8 - 7.43e7i)T + (5.99e16 - 1.03e17i)T^{2} \) |
| 83 | \( 1 + (4.43e8 - 4.43e8i)T - 1.86e17iT^{2} \) |
| 89 | \( 1 + (1.22e8 + 2.11e8i)T + (-1.75e17 + 3.03e17i)T^{2} \) |
| 97 | \( 1 + (2.41e8 + 2.41e8i)T + 7.60e17iT^{2} \) |
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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
−15.17723772365344191045537648757, −13.42266565363860214586777276399, −12.73185677438957531556503048491, −10.76639002960770678840220710206, −9.854436101209233506848184671720, −8.634951444435982480892397367312, −7.25411710539493408219215410862, −4.83700284285688483225277280635, −4.13919383948459952022609574966, −1.25289622479883233910106822369,
0.18238097262239200613356597412, 2.59900419298963991245512216978, 4.26678461086648059596991532393, 6.29603829620849446296934605453, 7.960803578145854985818461982975, 8.649915583836870290018975955231, 10.44449438382056499335149162437, 11.74111324083787724959180112604, 13.10859681175856006145688767985, 13.95674092484139566485880590760
