| L(s) = 1 | + (−3.64 + 6.30i)2-s + (−23.2 + 40.2i)3-s + (37.4 + 64.9i)4-s + 324.·5-s + (−169. − 292. i)6-s + (171.5 + 297. i)7-s − 1.47e3·8-s + (14.9 + 25.8i)9-s + (−1.18e3 + 2.04e3i)10-s + (−1.55e3 + 2.68e3i)11-s − 3.48e3·12-s + (7.29e3 − 3.09e3i)13-s − 2.49e3·14-s + (−7.54e3 + 1.30e4i)15-s + (587. − 1.01e3i)16-s + (1.61e4 + 2.80e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.321 + 0.557i)2-s + (−0.496 + 0.860i)3-s + (0.292 + 0.507i)4-s + 1.16·5-s + (−0.319 − 0.553i)6-s + (0.188 + 0.327i)7-s − 1.02·8-s + (0.00682 + 0.0118i)9-s + (−0.374 + 0.647i)10-s + (−0.351 + 0.608i)11-s − 0.581·12-s + (0.920 − 0.390i)13-s − 0.243·14-s + (−0.577 + 0.999i)15-s + (0.0358 − 0.0621i)16-s + (0.798 + 1.38i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.132i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.132i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.121014 - 1.81669i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.121014 - 1.81669i\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 + (-7.29e3 + 3.09e3i)T \) |
| good | 2 | \( 1 + (3.64 - 6.30i)T + (-64 - 110. i)T^{2} \) |
| 3 | \( 1 + (23.2 - 40.2i)T + (-1.09e3 - 1.89e3i)T^{2} \) |
| 5 | \( 1 - 324.T + 7.81e4T^{2} \) |
| 11 | \( 1 + (1.55e3 - 2.68e3i)T + (-9.74e6 - 1.68e7i)T^{2} \) |
| 17 | \( 1 + (-1.61e4 - 2.80e4i)T + (-2.05e8 + 3.55e8i)T^{2} \) |
| 19 | \( 1 + (-2.20e4 - 3.81e4i)T + (-4.46e8 + 7.74e8i)T^{2} \) |
| 23 | \( 1 + (-3.02e4 + 5.24e4i)T + (-1.70e9 - 2.94e9i)T^{2} \) |
| 29 | \( 1 + (7.32e4 - 1.26e5i)T + (-8.62e9 - 1.49e10i)T^{2} \) |
| 31 | \( 1 + 2.23e3T + 2.75e10T^{2} \) |
| 37 | \( 1 + (-1.63e5 + 2.83e5i)T + (-4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 + (5.50e4 - 9.53e4i)T + (-9.73e10 - 1.68e11i)T^{2} \) |
| 43 | \( 1 + (4.11e5 + 7.12e5i)T + (-1.35e11 + 2.35e11i)T^{2} \) |
| 47 | \( 1 - 1.29e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.66e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + (3.58e5 + 6.20e5i)T + (-1.24e12 + 2.15e12i)T^{2} \) |
| 61 | \( 1 + (1.27e6 + 2.21e6i)T + (-1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (1.94e6 - 3.37e6i)T + (-3.03e12 - 5.24e12i)T^{2} \) |
| 71 | \( 1 + (2.86e6 + 4.96e6i)T + (-4.54e12 + 7.87e12i)T^{2} \) |
| 73 | \( 1 - 1.91e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 1.86e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 3.77e5T + 2.71e13T^{2} \) |
| 89 | \( 1 + (-2.16e5 + 3.74e5i)T + (-2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 + (-1.91e6 - 3.31e6i)T + (-4.03e13 + 6.99e13i)T^{2} \) |
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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
−13.13052127329074774141291616291, −12.20483460821113536832134480335, −10.74236407501123572686975622685, −10.04089575890873994559143799416, −8.829726877310165789065703747773, −7.67010241618609796336329097135, −6.09081487014529044981661851449, −5.40529659080294110507409120501, −3.61425397809797392272266117572, −1.84811536094101374715143527964,
0.72271027483955501735299941951, 1.44181862898206583789749147624, 2.87243508074509935367387805117, 5.39478945704045967412766730141, 6.22197279561227675531965315348, 7.31549869606651401652916620367, 9.178254343418902870615411243096, 9.925570808906940560305208595019, 11.26541441425962619324799002100, 11.73817636470364872317569782402
