| L(s) = 1 | + (−9.79 + 5.65i)2-s + (63.9 − 110. i)4-s + (−935. − 539. i)5-s + (−2.05e3 − 3.56e3i)7-s + 1.44e3i·8-s + 1.22e4·10-s + (−8.41e3 + 4.86e3i)11-s + (−4.11e3 + 7.12e3i)13-s + (4.02e4 + 2.32e4i)14-s + (−8.19e3 − 1.41e4i)16-s − 8.78e4i·17-s + 1.23e5·19-s + (−1.19e5 + 6.91e4i)20-s + (5.49e4 − 9.52e4i)22-s + (−9.20e4 − 5.31e4i)23-s + ⋯ |
| L(s) = 1 | + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (−1.49 − 0.863i)5-s + (−0.856 − 1.48i)7-s + 0.353i·8-s + 1.22·10-s + (−0.575 + 0.332i)11-s + (−0.144 + 0.249i)13-s + (1.04 + 0.605i)14-s + (−0.125 − 0.216i)16-s − 1.05i·17-s + 0.948·19-s + (−0.748 + 0.431i)20-s + (0.234 − 0.406i)22-s + (−0.328 − 0.189i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0831 - 0.996i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.0831 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.0772932 + 0.0840134i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0772932 + 0.0840134i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (9.79 - 5.65i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (935. + 539. i)T + (1.95e5 + 3.38e5i)T^{2} \) |
| 7 | \( 1 + (2.05e3 + 3.56e3i)T + (-2.88e6 + 4.99e6i)T^{2} \) |
| 11 | \( 1 + (8.41e3 - 4.86e3i)T + (1.07e8 - 1.85e8i)T^{2} \) |
| 13 | \( 1 + (4.11e3 - 7.12e3i)T + (-4.07e8 - 7.06e8i)T^{2} \) |
| 17 | \( 1 + 8.78e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 - 1.23e5T + 1.69e10T^{2} \) |
| 23 | \( 1 + (9.20e4 + 5.31e4i)T + (3.91e10 + 6.78e10i)T^{2} \) |
| 29 | \( 1 + (4.30e5 - 2.48e5i)T + (2.50e11 - 4.33e11i)T^{2} \) |
| 31 | \( 1 + (5.77e5 - 1.00e6i)T + (-4.26e11 - 7.38e11i)T^{2} \) |
| 37 | \( 1 - 1.20e6T + 3.51e12T^{2} \) |
| 41 | \( 1 + (1.53e6 + 8.88e5i)T + (3.99e12 + 6.91e12i)T^{2} \) |
| 43 | \( 1 + (5.28e5 + 9.15e5i)T + (-5.84e12 + 1.01e13i)T^{2} \) |
| 47 | \( 1 + (-3.80e6 + 2.19e6i)T + (1.19e13 - 2.06e13i)T^{2} \) |
| 53 | \( 1 - 1.04e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 + (-8.70e6 - 5.02e6i)T + (7.34e13 + 1.27e14i)T^{2} \) |
| 61 | \( 1 + (4.15e6 + 7.20e6i)T + (-9.58e13 + 1.66e14i)T^{2} \) |
| 67 | \( 1 + (-6.19e6 + 1.07e7i)T + (-2.03e14 - 3.51e14i)T^{2} \) |
| 71 | \( 1 + 1.69e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 2.42e7T + 8.06e14T^{2} \) |
| 79 | \( 1 + (-6.39e6 - 1.10e7i)T + (-7.58e14 + 1.31e15i)T^{2} \) |
| 83 | \( 1 + (-2.79e7 + 1.61e7i)T + (1.12e15 - 1.95e15i)T^{2} \) |
| 89 | \( 1 - 2.10e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + (4.46e7 + 7.73e7i)T + (-3.91e15 + 6.78e15i)T^{2} \) |
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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
−13.94791508357164751457412533200, −12.73147599799466635977040780897, −11.56041509527600352982917540268, −10.30421691962491193689332732943, −9.093638536652161897280873365534, −7.65379282099176829755631208607, −7.11621563662287317790527700932, −4.89739888234014492795718253223, −3.56744853127841530059216060878, −0.824562574835914713448161322089,
0.07081448328502569536036221342, 2.60345446034590085924713752280, 3.62108553066338800218838226997, 5.95127481889391926694940600256, 7.48122468247397366907322326125, 8.455656191040200977126606165534, 9.817482655272960012789299480463, 11.14038629623029441006306454949, 11.93806505665304700531268140210, 12.92704122499152963205838582142
