| L(s) = 1 | + (1.46 − 0.944i)2-s + (3.84 − 26.7i)3-s + (−51.9 + 113. i)4-s + (−216. + 63.6i)5-s + (−19.5 − 42.9i)6-s + (397. + 458. i)7-s + (62.8 + 437. i)8-s + (−699. − 205. i)9-s + (−258. + 298. i)10-s + (−1.24e3 − 800. i)11-s + (2.83e3 + 1.82e3i)12-s + (5.86e3 − 6.76e3i)13-s + (1.01e3 + 298. i)14-s + (868. + 6.03e3i)15-s + (−9.96e3 − 1.15e4i)16-s + (−4.63e3 − 1.01e4i)17-s + ⋯ |
| L(s) = 1 | + (0.129 − 0.0834i)2-s + (0.0821 − 0.571i)3-s + (−0.405 + 0.887i)4-s + (−0.775 + 0.227i)5-s + (−0.0370 − 0.0811i)6-s + (0.437 + 0.505i)7-s + (0.0434 + 0.302i)8-s + (−0.319 − 0.0939i)9-s + (−0.0817 + 0.0943i)10-s + (−0.281 − 0.181i)11-s + (0.474 + 0.304i)12-s + (0.739 − 0.853i)13-s + (0.0990 + 0.0290i)14-s + (0.0664 + 0.461i)15-s + (−0.608 − 0.702i)16-s + (−0.228 − 0.500i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.256 + 0.966i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.256 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.575088 - 0.747536i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.575088 - 0.747536i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-3.84 + 26.7i)T \) |
| 23 | \( 1 + (2.82e4 - 5.10e4i)T \) |
| good | 2 | \( 1 + (-1.46 + 0.944i)T + (53.1 - 116. i)T^{2} \) |
| 5 | \( 1 + (216. - 63.6i)T + (6.57e4 - 4.22e4i)T^{2} \) |
| 7 | \( 1 + (-397. - 458. i)T + (-1.17e5 + 8.15e5i)T^{2} \) |
| 11 | \( 1 + (1.24e3 + 800. i)T + (8.09e6 + 1.77e7i)T^{2} \) |
| 13 | \( 1 + (-5.86e3 + 6.76e3i)T + (-8.93e6 - 6.21e7i)T^{2} \) |
| 17 | \( 1 + (4.63e3 + 1.01e4i)T + (-2.68e8 + 3.10e8i)T^{2} \) |
| 19 | \( 1 + (-1.66e4 + 3.64e4i)T + (-5.85e8 - 6.75e8i)T^{2} \) |
| 29 | \( 1 + (2.59e4 + 5.67e4i)T + (-1.12e10 + 1.30e10i)T^{2} \) |
| 31 | \( 1 + (4.26e4 + 2.96e5i)T + (-2.63e10 + 7.75e9i)T^{2} \) |
| 37 | \( 1 + (1.18e4 + 3.48e3i)T + (7.98e10 + 5.13e10i)T^{2} \) |
| 41 | \( 1 + (-5.95e5 + 1.74e5i)T + (1.63e11 - 1.05e11i)T^{2} \) |
| 43 | \( 1 + (-7.41e4 + 5.15e5i)T + (-2.60e11 - 7.65e10i)T^{2} \) |
| 47 | \( 1 - 1.16e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + (7.36e5 + 8.49e5i)T + (-1.67e11 + 1.16e12i)T^{2} \) |
| 59 | \( 1 + (6.55e5 - 7.56e5i)T + (-3.54e11 - 2.46e12i)T^{2} \) |
| 61 | \( 1 + (-2.14e5 - 1.49e6i)T + (-3.01e12 + 8.85e11i)T^{2} \) |
| 67 | \( 1 + (-1.35e6 + 8.73e5i)T + (2.51e12 - 5.51e12i)T^{2} \) |
| 71 | \( 1 + (2.23e6 - 1.43e6i)T + (3.77e12 - 8.27e12i)T^{2} \) |
| 73 | \( 1 + (1.45e6 - 3.19e6i)T + (-7.23e12 - 8.34e12i)T^{2} \) |
| 79 | \( 1 + (-2.94e6 + 3.40e6i)T + (-2.73e12 - 1.90e13i)T^{2} \) |
| 83 | \( 1 + (-1.33e5 - 3.91e4i)T + (2.28e13 + 1.46e13i)T^{2} \) |
| 89 | \( 1 + (9.77e5 - 6.79e6i)T + (-4.24e13 - 1.24e13i)T^{2} \) |
| 97 | \( 1 + (-5.00e6 + 1.46e6i)T + (6.79e13 - 4.36e13i)T^{2} \) |
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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
−13.04778642325150904394427471072, −11.78633387172695355027073337606, −11.24819702411232579973766900086, −9.194471177092302602317732200801, −8.059165549754712563887095212835, −7.39673828057190711978479593885, −5.51626965537564375030223457974, −3.86848719943290367894674113109, −2.58782627518484954202453516964, −0.34072257982216752746774283961,
1.37660134993160667117609259993, 3.87820122612692673551076403462, 4.73418437110855369299508951808, 6.20947144607388572309274996158, 7.916096657771394314499665116098, 9.061282345181875937165108189157, 10.32246873944759749283237246136, 11.13018005351975699235860816618, 12.50666378141684306093705344019, 14.01778505321118458239290980092
