| L(s) = 1 | + (−40.1 + 69.5i)3-s + (−97.7 + 169. i)5-s − 587.·7-s + (−2.13e3 − 3.69e3i)9-s − 6.54e3·11-s + (2.81e3 + 4.87e3i)13-s + (−7.85e3 − 1.36e4i)15-s + (−1.12e4 + 1.94e4i)17-s + (2.98e4 + 169. i)19-s + (2.36e4 − 4.08e4i)21-s + (−4.29e4 − 7.43e4i)23-s + (1.99e4 + 3.45e4i)25-s + 1.67e5·27-s + (7.43e4 + 1.28e5i)29-s + 1.78e5·31-s + ⋯ |
| L(s) = 1 | + (−0.858 + 1.48i)3-s + (−0.349 + 0.605i)5-s − 0.647·7-s + (−0.975 − 1.69i)9-s − 1.48·11-s + (0.355 + 0.614i)13-s + (−0.600 − 1.04i)15-s + (−0.555 + 0.962i)17-s + (0.999 + 0.00567i)19-s + (0.556 − 0.963i)21-s + (−0.735 − 1.27i)23-s + (0.255 + 0.442i)25-s + 1.63·27-s + (0.566 + 0.980i)29-s + 1.07·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.667 + 0.744i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.667 + 0.744i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.0536244 - 0.0239477i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0536244 - 0.0239477i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (-2.98e4 - 169. i)T \) |
| good | 3 | \( 1 + (40.1 - 69.5i)T + (-1.09e3 - 1.89e3i)T^{2} \) |
| 5 | \( 1 + (97.7 - 169. i)T + (-3.90e4 - 6.76e4i)T^{2} \) |
| 7 | \( 1 + 587.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 6.54e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + (-2.81e3 - 4.87e3i)T + (-3.13e7 + 5.43e7i)T^{2} \) |
| 17 | \( 1 + (1.12e4 - 1.94e4i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 23 | \( 1 + (4.29e4 + 7.43e4i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 + (-7.43e4 - 1.28e5i)T + (-8.62e9 + 1.49e10i)T^{2} \) |
| 31 | \( 1 - 1.78e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.11e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (-1.12e5 + 1.94e5i)T + (-9.73e10 - 1.68e11i)T^{2} \) |
| 43 | \( 1 + (-7.83e4 + 1.35e5i)T + (-1.35e11 - 2.35e11i)T^{2} \) |
| 47 | \( 1 + (3.35e5 + 5.81e5i)T + (-2.53e11 + 4.38e11i)T^{2} \) |
| 53 | \( 1 + (-7.18e5 - 1.24e6i)T + (-5.87e11 + 1.01e12i)T^{2} \) |
| 59 | \( 1 + (-1.19e6 + 2.07e6i)T + (-1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (1.18e4 + 2.04e4i)T + (-1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-2.12e5 - 3.67e5i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 + (-6.35e5 + 1.10e6i)T + (-4.54e12 - 7.87e12i)T^{2} \) |
| 73 | \( 1 + (3.00e6 - 5.20e6i)T + (-5.52e12 - 9.56e12i)T^{2} \) |
| 79 | \( 1 + (1.79e6 - 3.11e6i)T + (-9.60e12 - 1.66e13i)T^{2} \) |
| 83 | \( 1 + 8.04e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + (4.96e6 + 8.60e6i)T + (-2.21e13 + 3.83e13i)T^{2} \) |
| 97 | \( 1 + (-5.27e6 + 9.13e6i)T + (-4.03e13 - 6.99e13i)T^{2} \) |
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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
−12.74116787835888411202920746173, −11.46997876045287875097844572792, −10.56084512466149962448381738229, −10.00225254159367222613946100456, −8.559178152657997257518858732116, −6.74903690091635178218900073655, −5.52643596080158033277416848308, −4.27756898558710626291150489455, −3.04001480523338638553821147582, −0.02908084553215729093523946877,
0.915280827300227409606866174460, 2.69686973105395832132703089488, 5.06539370773075605138978392529, 6.10537375132824015310573106355, 7.41406294315765001167584077105, 8.191378804513886024268508035055, 9.992665795932852125261160611582, 11.41376567024851047343478707268, 12.18467317268370108240583093004, 13.21058805488677264026284350380
