| L(s) = 1 | + (−3.21 − 3.71i)2-s + (7.57 + 4.86i)3-s + (1.11 − 7.77i)4-s + (37.1 − 81.2i)5-s + (−6.29 − 43.7i)6-s + (196. + 57.6i)7-s + (−164. + 105. i)8-s + (33.6 + 73.6i)9-s + (−421. + 123. i)10-s + (48.2 − 55.6i)11-s + (46.2 − 53.4i)12-s + (305. − 89.6i)13-s + (−417. − 913. i)14-s + (676. − 434. i)15-s + (682. + 200. i)16-s + (−13.2 − 92.4i)17-s + ⋯ |
| L(s) = 1 | + (−0.568 − 0.656i)2-s + (0.485 + 0.312i)3-s + (0.0349 − 0.242i)4-s + (0.663 − 1.45i)5-s + (−0.0713 − 0.496i)6-s + (1.51 + 0.444i)7-s + (−0.910 + 0.584i)8-s + (0.138 + 0.303i)9-s + (−1.33 + 0.391i)10-s + (0.120 − 0.138i)11-s + (0.0928 − 0.107i)12-s + (0.501 − 0.147i)13-s + (−0.569 − 1.24i)14-s + (0.776 − 0.498i)15-s + (0.666 + 0.195i)16-s + (−0.0111 − 0.0775i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.259 + 0.965i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.259 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.15812 - 1.50986i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.15812 - 1.50986i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-7.57 - 4.86i)T \) |
| 23 | \( 1 + (1.68e3 + 1.89e3i)T \) |
| good | 2 | \( 1 + (3.21 + 3.71i)T + (-4.55 + 31.6i)T^{2} \) |
| 5 | \( 1 + (-37.1 + 81.2i)T + (-2.04e3 - 2.36e3i)T^{2} \) |
| 7 | \( 1 + (-196. - 57.6i)T + (1.41e4 + 9.08e3i)T^{2} \) |
| 11 | \( 1 + (-48.2 + 55.6i)T + (-2.29e4 - 1.59e5i)T^{2} \) |
| 13 | \( 1 + (-305. + 89.6i)T + (3.12e5 - 2.00e5i)T^{2} \) |
| 17 | \( 1 + (13.2 + 92.4i)T + (-1.36e6 + 4.00e5i)T^{2} \) |
| 19 | \( 1 + (-157. + 1.09e3i)T + (-2.37e6 - 6.97e5i)T^{2} \) |
| 29 | \( 1 + (647. + 4.50e3i)T + (-1.96e7 + 5.77e6i)T^{2} \) |
| 31 | \( 1 + (5.72e3 - 3.67e3i)T + (1.18e7 - 2.60e7i)T^{2} \) |
| 37 | \( 1 + (-6.61e3 - 1.44e4i)T + (-4.54e7 + 5.24e7i)T^{2} \) |
| 41 | \( 1 + (-3.51e3 + 7.70e3i)T + (-7.58e7 - 8.75e7i)T^{2} \) |
| 43 | \( 1 + (3.00e3 + 1.93e3i)T + (6.10e7 + 1.33e8i)T^{2} \) |
| 47 | \( 1 + 1.32e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + (-3.07e4 - 9.02e3i)T + (3.51e8 + 2.26e8i)T^{2} \) |
| 59 | \( 1 + (2.00e4 - 5.89e3i)T + (6.01e8 - 3.86e8i)T^{2} \) |
| 61 | \( 1 + (-2.12e3 + 1.36e3i)T + (3.50e8 - 7.68e8i)T^{2} \) |
| 67 | \( 1 + (-1.23e4 - 1.42e4i)T + (-1.92e8 + 1.33e9i)T^{2} \) |
| 71 | \( 1 + (1.39e4 + 1.61e4i)T + (-2.56e8 + 1.78e9i)T^{2} \) |
| 73 | \( 1 + (1.07e4 - 7.45e4i)T + (-1.98e9 - 5.84e8i)T^{2} \) |
| 79 | \( 1 + (3.36e4 - 9.89e3i)T + (2.58e9 - 1.66e9i)T^{2} \) |
| 83 | \( 1 + (-3.80e4 - 8.32e4i)T + (-2.57e9 + 2.97e9i)T^{2} \) |
| 89 | \( 1 + (-1.03e5 - 6.67e4i)T + (2.31e9 + 5.07e9i)T^{2} \) |
| 97 | \( 1 + (-6.22e4 + 1.36e5i)T + (-5.62e9 - 6.48e9i)T^{2} \) |
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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
−13.45131063457254234852305584288, −12.08400695445613450916089654712, −11.10226385818363214529364382388, −9.848611155881675848206003656449, −8.809590103219905677065841111164, −8.312785687355902977424347779071, −5.68283134455991329254626481221, −4.67875313285405605906607425905, −2.16103080810466519964002905178, −1.08945074506670663979761682861,
1.89924246076749951806542784663, 3.62097267349560583461263157317, 6.01837208945016922038372887395, 7.27777816557497282726961597157, 7.904205482456319251820991634273, 9.270237444320433220871832398709, 10.61690798720745325574275257695, 11.65567251713159234409829153900, 13.28904414632968917619505635501, 14.55365384925459532384801178014
