| L(s) = 1 | + (−9.68 − 9.68i)3-s + (−48.6 + 48.6i)7-s − 55.4i·9-s − 463. i·11-s + (−320. + 320. i)13-s + (−1.04e3 − 1.04e3i)17-s + 701.·19-s + 942.·21-s + (2.00e3 + 2.00e3i)23-s + (−2.89e3 + 2.89e3i)27-s − 3.56e3i·29-s − 9.04e3i·31-s + (−4.48e3 + 4.48e3i)33-s + (−1.64e3 − 1.64e3i)37-s + 6.21e3·39-s + ⋯ |
| L(s) = 1 | + (−0.621 − 0.621i)3-s + (−0.375 + 0.375i)7-s − 0.228i·9-s − 1.15i·11-s + (−0.526 + 0.526i)13-s + (−0.877 − 0.877i)17-s + 0.445·19-s + 0.466·21-s + (0.789 + 0.789i)23-s + (−0.762 + 0.762i)27-s − 0.787i·29-s − 1.69i·31-s + (−0.716 + 0.716i)33-s + (−0.197 − 0.197i)37-s + 0.654·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0299 - 0.999i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.0299 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.2370155592\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2370155592\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + (9.68 + 9.68i)T + 243iT^{2} \) |
| 7 | \( 1 + (48.6 - 48.6i)T - 1.68e4iT^{2} \) |
| 11 | \( 1 + 463. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + (320. - 320. i)T - 3.71e5iT^{2} \) |
| 17 | \( 1 + (1.04e3 + 1.04e3i)T + 1.41e6iT^{2} \) |
| 19 | \( 1 - 701.T + 2.47e6T^{2} \) |
| 23 | \( 1 + (-2.00e3 - 2.00e3i)T + 6.43e6iT^{2} \) |
| 29 | \( 1 + 3.56e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 9.04e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 + (1.64e3 + 1.64e3i)T + 6.93e7iT^{2} \) |
| 41 | \( 1 + 1.43e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + (3.94e3 + 3.94e3i)T + 1.47e8iT^{2} \) |
| 47 | \( 1 + (-7.94e3 + 7.94e3i)T - 2.29e8iT^{2} \) |
| 53 | \( 1 + (1.16e4 - 1.16e4i)T - 4.18e8iT^{2} \) |
| 59 | \( 1 - 1.12e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.93e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + (9.19e3 - 9.19e3i)T - 1.35e9iT^{2} \) |
| 71 | \( 1 - 5.26e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + (-2.79e4 + 2.79e4i)T - 2.07e9iT^{2} \) |
| 79 | \( 1 - 8.22e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + (-7.72e4 - 7.72e4i)T + 3.93e9iT^{2} \) |
| 89 | \( 1 - 1.45e5iT - 5.58e9T^{2} \) |
| 97 | \( 1 + (9.78e4 + 9.78e4i)T + 8.58e9iT^{2} \) |
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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
−11.01032936638086082720048650800, −9.563496474117617843208201838293, −9.038100217926966984228181696790, −7.73056034053612159205490631299, −6.78964890685026673429720656745, −6.04164656198451446306162582423, −5.10833627640420157976898942744, −3.63772613382884326267815685381, −2.40648958338761046782658472406, −0.893142287553534978706838897200,
0.07822291184859071298741930727, 1.78610768785169913237148160106, 3.25485508457461116359489517593, 4.60659900240237908559999971580, 5.09899941351787725735893517284, 6.48221193637887301791063171290, 7.27408510626926612392301975853, 8.456623153565869282628283147098, 9.566546742486243429767364611568, 10.44372011930493581420413062657
