| L(s) = 1 | + (10.4 − 18.0i)2-s + (−36.8 + 63.8i)3-s + (−153. − 265. i)4-s + 497.·5-s + (768. + 1.33e3i)6-s + (171.5 + 297. i)7-s − 3.71e3·8-s + (−1.62e3 − 2.81e3i)9-s + (5.18e3 − 8.98e3i)10-s + (−3.77e3 + 6.53e3i)11-s + 2.25e4·12-s + (−6.43e3 + 4.62e3i)13-s + 7.14e3·14-s + (−1.83e4 + 3.17e4i)15-s + (−1.90e4 + 3.30e4i)16-s + (1.97e3 + 3.42e3i)17-s + ⋯ |
| L(s) = 1 | + (0.920 − 1.59i)2-s + (−0.788 + 1.36i)3-s + (−1.19 − 2.07i)4-s + 1.78·5-s + (1.45 + 2.51i)6-s + (0.188 + 0.327i)7-s − 2.56·8-s + (−0.744 − 1.28i)9-s + (1.63 − 2.84i)10-s + (−0.855 + 1.48i)11-s + 3.77·12-s + (−0.812 + 0.583i)13-s + 0.696·14-s + (−1.40 + 2.43i)15-s + (−1.16 + 2.01i)16-s + (0.0975 + 0.168i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.937 - 0.347i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.937 - 0.347i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.38043 + 0.426447i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.38043 + 0.426447i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-171.5 - 297. i)T \) |
| 13 | \( 1 + (6.43e3 - 4.62e3i)T \) |
| good | 2 | \( 1 + (-10.4 + 18.0i)T + (-64 - 110. i)T^{2} \) |
| 3 | \( 1 + (36.8 - 63.8i)T + (-1.09e3 - 1.89e3i)T^{2} \) |
| 5 | \( 1 - 497.T + 7.81e4T^{2} \) |
| 11 | \( 1 + (3.77e3 - 6.53e3i)T + (-9.74e6 - 1.68e7i)T^{2} \) |
| 17 | \( 1 + (-1.97e3 - 3.42e3i)T + (-2.05e8 + 3.55e8i)T^{2} \) |
| 19 | \( 1 + (-1.30e4 - 2.26e4i)T + (-4.46e8 + 7.74e8i)T^{2} \) |
| 23 | \( 1 + (5.73e3 - 9.94e3i)T + (-1.70e9 - 2.94e9i)T^{2} \) |
| 29 | \( 1 + (-2.88e4 + 5.00e4i)T + (-8.62e9 - 1.49e10i)T^{2} \) |
| 31 | \( 1 - 2.30e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + (2.06e5 - 3.57e5i)T + (-4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 + (3.27e5 - 5.66e5i)T + (-9.73e10 - 1.68e11i)T^{2} \) |
| 43 | \( 1 + (-1.99e5 - 3.44e5i)T + (-1.35e11 + 2.35e11i)T^{2} \) |
| 47 | \( 1 - 9.59e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 6.80e4T + 1.17e12T^{2} \) |
| 59 | \( 1 + (-3.15e5 - 5.46e5i)T + (-1.24e12 + 2.15e12i)T^{2} \) |
| 61 | \( 1 + (6.12e5 + 1.06e6i)T + (-1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-6.17e5 + 1.06e6i)T + (-3.03e12 - 5.24e12i)T^{2} \) |
| 71 | \( 1 + (6.18e5 + 1.07e6i)T + (-4.54e12 + 7.87e12i)T^{2} \) |
| 73 | \( 1 + 2.14e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 6.17e5T + 1.92e13T^{2} \) |
| 83 | \( 1 + 1.82e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + (-3.53e6 + 6.11e6i)T + (-2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 + (3.72e6 + 6.44e6i)T + (-4.03e13 + 6.99e13i)T^{2} \) |
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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
−12.50185825224519428955536060151, −11.68055320220783697237080140056, −10.26624930620971597730422469321, −10.08277309594987220506169394230, −9.415026740961600301246672753972, −6.06994030209165792825498723986, −5.07005490811650815202594331953, −4.59117994037903759433447683623, −2.72857020732265904406900243313, −1.65941357080398570014142095289,
0.61067033592531181698410713095, 2.63937269470881206724446875886, 5.29594617467450620209636862756, 5.62493572653138792354322993563, 6.62842188979691069619180318122, 7.49179833465063812675775217993, 8.723905050735506231253120448111, 10.55082537266722425570584626115, 12.21410225300491691831506764412, 13.15403793701958152504935322702
