L(s) = 1 | + (41.8 − 17.1i)2-s − 243·3-s + (1.45e3 − 1.43e3i)4-s − 1.03e4i·5-s + (−1.01e4 + 4.17e3i)6-s + (−1.95e4 − 3.99e4i)7-s + (3.62e4 − 8.53e4i)8-s + 5.90e4·9-s + (−1.78e5 − 4.33e5i)10-s − 9.48e5i·11-s + (−3.54e5 + 3.49e5i)12-s − 4.11e5i·13-s + (−1.50e6 − 1.33e6i)14-s + 2.51e6i·15-s + (5.06e4 − 4.19e6i)16-s + 1.70e6i·17-s + ⋯ |
L(s) = 1 | + (0.925 − 0.379i)2-s − 0.577·3-s + (0.711 − 0.702i)4-s − 1.48i·5-s + (−0.534 + 0.219i)6-s + (−0.440 − 0.897i)7-s + (0.391 − 0.920i)8-s + 0.333·9-s + (−0.563 − 1.37i)10-s − 1.77i·11-s + (−0.410 + 0.405i)12-s − 0.307i·13-s + (−0.748 − 0.663i)14-s + 0.856i·15-s + (0.0120 − 0.999i)16-s + 0.291i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 - 0.329i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.944 - 0.329i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(6)\) |
\(\approx\) |
\(0.502692 + 2.97052i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.502692 + 2.97052i\) |
\(L(\frac{13}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-41.8 + 17.1i)T \) |
| 3 | \( 1 + 243T \) |
| 7 | \( 1 + (1.95e4 + 3.99e4i)T \) |
good | 5 | \( 1 + 1.03e4iT - 4.88e7T^{2} \) |
| 11 | \( 1 + 9.48e5iT - 2.85e11T^{2} \) |
| 13 | \( 1 + 4.11e5iT - 1.79e12T^{2} \) |
| 17 | \( 1 - 1.70e6iT - 3.42e13T^{2} \) |
| 19 | \( 1 - 1.17e7T + 1.16e14T^{2} \) |
| 23 | \( 1 + 4.13e6iT - 9.52e14T^{2} \) |
| 29 | \( 1 - 1.88e8T + 1.22e16T^{2} \) |
| 31 | \( 1 + 1.51e8T + 2.54e16T^{2} \) |
| 37 | \( 1 - 6.81e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 8.50e8iT - 5.50e17T^{2} \) |
| 43 | \( 1 + 1.82e8iT - 9.29e17T^{2} \) |
| 47 | \( 1 - 1.56e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 1.37e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + 5.42e9T + 3.01e19T^{2} \) |
| 61 | \( 1 + 8.36e9iT - 4.35e19T^{2} \) |
| 67 | \( 1 + 4.94e9iT - 1.22e20T^{2} \) |
| 71 | \( 1 - 1.73e10iT - 2.31e20T^{2} \) |
| 73 | \( 1 - 1.99e10iT - 3.13e20T^{2} \) |
| 79 | \( 1 - 1.02e10iT - 7.47e20T^{2} \) |
| 83 | \( 1 - 5.29e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 8.39e10iT - 2.77e21T^{2} \) |
| 97 | \( 1 - 6.83e10iT - 7.15e21T^{2} \) |
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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
−11.58274645015460035380638749885, −10.65239261511953457804180444662, −9.443850621103679321721182702974, −7.961595587931932115465101911952, −6.34032017511294233516774070154, −5.41643389332820304426661901343, −4.37736463656660390774402287375, −3.21012737789209096598777054967, −1.05460571214358940562420540020, −0.68207489069887558740317899686,
2.11903046281526910145443746665, 3.07900184215445540093329639080, 4.55879221576959431114172278701, 5.82627135149747122341684081520, 6.81252861109160013626505387405, 7.48357879802960884405206505934, 9.553234027967577111920329347620, 10.70334085406455556178069109374, 11.84927887605993118570103127852, 12.42022662227932181389650569810