| L(s) = 1 | + (−10.9 + 7.01i)2-s + (−2.21 + 15.4i)3-s + (43.4 − 95.0i)4-s + (−4.24 − 14.4i)5-s + (−84.0 − 184. i)6-s + (−345. + 298. i)7-s + (74.8 + 520. i)8-s + (−233. − 68.4i)9-s + (147. + 128. i)10-s + (1.08e3 − 1.69e3i)11-s + (1.37e3 + 880. i)12-s + (2.29e3 − 2.65e3i)13-s + (1.66e3 − 5.68e3i)14-s + (232. − 33.4i)15-s + (−92.1 − 106. i)16-s + (−5.46e3 + 2.49e3i)17-s + ⋯ |
| L(s) = 1 | + (−1.36 + 0.877i)2-s + (−0.0821 + 0.571i)3-s + (0.678 − 1.48i)4-s + (−0.0339 − 0.115i)5-s + (−0.389 − 0.852i)6-s + (−1.00 + 0.871i)7-s + (0.146 + 1.01i)8-s + (−0.319 − 0.0939i)9-s + (0.147 + 0.128i)10-s + (0.816 − 1.27i)11-s + (0.793 + 0.509i)12-s + (1.04 − 1.20i)13-s + (0.608 − 2.07i)14-s + (0.0688 − 0.00990i)15-s + (−0.0225 − 0.0259i)16-s + (−1.11 + 0.508i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.288 - 0.957i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.288 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.578228 + 0.429834i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.578228 + 0.429834i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.21 - 15.4i)T \) |
| 23 | \( 1 + (1.06e4 + 5.84e3i)T \) |
| good | 2 | \( 1 + (10.9 - 7.01i)T + (26.5 - 58.2i)T^{2} \) |
| 5 | \( 1 + (4.24 + 14.4i)T + (-1.31e4 + 8.44e3i)T^{2} \) |
| 7 | \( 1 + (345. - 298. i)T + (1.67e4 - 1.16e5i)T^{2} \) |
| 11 | \( 1 + (-1.08e3 + 1.69e3i)T + (-7.35e5 - 1.61e6i)T^{2} \) |
| 13 | \( 1 + (-2.29e3 + 2.65e3i)T + (-6.86e5 - 4.77e6i)T^{2} \) |
| 17 | \( 1 + (5.46e3 - 2.49e3i)T + (1.58e7 - 1.82e7i)T^{2} \) |
| 19 | \( 1 + (-6.64e3 - 3.03e3i)T + (3.08e7 + 3.55e7i)T^{2} \) |
| 29 | \( 1 + (666. + 1.46e3i)T + (-3.89e8 + 4.49e8i)T^{2} \) |
| 31 | \( 1 + (-6.30e3 - 4.38e4i)T + (-8.51e8 + 2.50e8i)T^{2} \) |
| 37 | \( 1 + (1.67e4 - 5.69e4i)T + (-2.15e9 - 1.38e9i)T^{2} \) |
| 41 | \( 1 + (-6.34e4 + 1.86e4i)T + (3.99e9 - 2.56e9i)T^{2} \) |
| 43 | \( 1 + (2.03e4 + 2.93e3i)T + (6.06e9 + 1.78e9i)T^{2} \) |
| 47 | \( 1 - 1.27e5T + 1.07e10T^{2} \) |
| 53 | \( 1 + (1.23e5 - 1.07e5i)T + (3.15e9 - 2.19e10i)T^{2} \) |
| 59 | \( 1 + (-1.40e5 + 1.61e5i)T + (-6.00e9 - 4.17e10i)T^{2} \) |
| 61 | \( 1 + (-1.26e5 + 1.82e4i)T + (4.94e10 - 1.45e10i)T^{2} \) |
| 67 | \( 1 + (-1.85e5 - 2.88e5i)T + (-3.75e10 + 8.22e10i)T^{2} \) |
| 71 | \( 1 + (-3.26e5 + 2.09e5i)T + (5.32e10 - 1.16e11i)T^{2} \) |
| 73 | \( 1 + (-5.59e4 + 1.22e5i)T + (-9.91e10 - 1.14e11i)T^{2} \) |
| 79 | \( 1 + (-3.31e5 - 2.87e5i)T + (3.45e10 + 2.40e11i)T^{2} \) |
| 83 | \( 1 + (-5.67e4 + 1.93e5i)T + (-2.75e11 - 1.76e11i)T^{2} \) |
| 89 | \( 1 + (5.24e5 + 7.53e4i)T + (4.76e11 + 1.40e11i)T^{2} \) |
| 97 | \( 1 + (4.03e5 + 1.37e6i)T + (-7.00e11 + 4.50e11i)T^{2} \) |
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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
−14.08380855757486538846042342959, −12.54930708528754938882909800515, −11.00470734035420456073896481230, −10.01822836815151539824941949367, −8.826405887893175774931853475435, −8.422218306490267855373074300473, −6.46365068136437849061722905833, −5.81031491878359247344086594055, −3.36141691469121639518167436022, −0.74573130043680759105213313323,
0.77294791017482965317228811516, 2.10096444765249016624675355622, 3.87935750210742545385626679416, 6.66442986952523561625841443447, 7.41607472961785022092334552845, 9.107022747724072695912767746211, 9.665019662201259884112381561447, 11.03764123468345563426025105656, 11.79728856041189211341434124783, 13.01888719674365446558896628244
