| L(s) = 1 | + (84.9 − 30.9i)3-s + (−77.9 + 442. i)5-s + (−193. − 334. i)7-s + (4.57e3 − 3.84e3i)9-s + (−1.89e3 + 3.28e3i)11-s + (9.07e3 + 3.30e3i)13-s + (7.04e3 + 3.99e4i)15-s + (2.18e4 + 1.83e4i)17-s + (4.46e3 − 2.95e4i)19-s + (−2.67e4 − 2.24e4i)21-s + (1.06e4 + 6.06e4i)23-s + (−1.16e5 − 4.22e4i)25-s + (1.71e5 − 2.96e5i)27-s + (3.21e4 − 2.70e4i)29-s + (9.24e4 + 1.60e5i)31-s + ⋯ |
| L(s) = 1 | + (1.81 − 0.660i)3-s + (−0.278 + 1.58i)5-s + (−0.212 − 0.368i)7-s + (2.09 − 1.75i)9-s + (−0.430 + 0.744i)11-s + (1.14 + 0.417i)13-s + (0.538 + 3.05i)15-s + (1.07 + 0.905i)17-s + (0.149 − 0.988i)19-s + (−0.630 − 0.528i)21-s + (0.183 + 1.03i)23-s + (−1.48 − 0.540i)25-s + (1.67 − 2.89i)27-s + (0.245 − 0.205i)29-s + (0.557 + 0.965i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 - 0.379i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.925 - 0.379i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(3.55659 + 0.700607i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.55659 + 0.700607i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (-4.46e3 + 2.95e4i)T \) |
| good | 3 | \( 1 + (-84.9 + 30.9i)T + (1.67e3 - 1.40e3i)T^{2} \) |
| 5 | \( 1 + (77.9 - 442. i)T + (-7.34e4 - 2.67e4i)T^{2} \) |
| 7 | \( 1 + (193. + 334. i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + (1.89e3 - 3.28e3i)T + (-9.74e6 - 1.68e7i)T^{2} \) |
| 13 | \( 1 + (-9.07e3 - 3.30e3i)T + (4.80e7 + 4.03e7i)T^{2} \) |
| 17 | \( 1 + (-2.18e4 - 1.83e4i)T + (7.12e7 + 4.04e8i)T^{2} \) |
| 23 | \( 1 + (-1.06e4 - 6.06e4i)T + (-3.19e9 + 1.16e9i)T^{2} \) |
| 29 | \( 1 + (-3.21e4 + 2.70e4i)T + (2.99e9 - 1.69e10i)T^{2} \) |
| 31 | \( 1 + (-9.24e4 - 1.60e5i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 + 3.54e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (2.01e5 - 7.32e4i)T + (1.49e11 - 1.25e11i)T^{2} \) |
| 43 | \( 1 + (-4.79e4 + 2.72e5i)T + (-2.55e11 - 9.29e10i)T^{2} \) |
| 47 | \( 1 + (-8.47e4 + 7.10e4i)T + (8.79e10 - 4.98e11i)T^{2} \) |
| 53 | \( 1 + (-1.69e5 - 9.64e5i)T + (-1.10e12 + 4.01e11i)T^{2} \) |
| 59 | \( 1 + (9.54e5 + 8.00e5i)T + (4.32e11 + 2.45e12i)T^{2} \) |
| 61 | \( 1 + (6.11e5 + 3.46e6i)T + (-2.95e12 + 1.07e12i)T^{2} \) |
| 67 | \( 1 + (3.07e5 - 2.58e5i)T + (1.05e12 - 5.96e12i)T^{2} \) |
| 71 | \( 1 + (-2.05e5 + 1.16e6i)T + (-8.54e12 - 3.11e12i)T^{2} \) |
| 73 | \( 1 + (1.06e6 - 3.87e5i)T + (8.46e12 - 7.10e12i)T^{2} \) |
| 79 | \( 1 + (-3.82e6 + 1.39e6i)T + (1.47e13 - 1.23e13i)T^{2} \) |
| 83 | \( 1 + (-2.42e6 - 4.20e6i)T + (-1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 + (8.07e6 + 2.94e6i)T + (3.38e13 + 2.84e13i)T^{2} \) |
| 97 | \( 1 + (1.01e7 + 8.50e6i)T + (1.40e13 + 7.95e13i)T^{2} \) |
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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
−13.59716724756969116426795208002, −12.30051077355421476948040009118, −10.69636051537665999489659030098, −9.689175418345295817028012764414, −8.341846103188570845691112345726, −7.33434304820150180096385756089, −6.66143181735953857761631921665, −3.72557880051428745320984523779, −3.01173899341403440056204759858, −1.63968835861335736919827137563,
1.16129361511880253413701175819, 2.94463806601168128390894180586, 4.07553554107132040506006686415, 5.39165673525370814627044195969, 7.920548411854379323667813842436, 8.481254364565908067376004758731, 9.256919831992216604215921789141, 10.37311923947747267959630388738, 12.23963884327991572349420292724, 13.26326352433169183540053562440
