L(s) = 1 | − 50.6i·3-s + 207.·5-s − 216.·7-s − 1.83e3·9-s − 1.26e3·11-s + 2.82e3i·13-s − 1.05e4i·15-s + 3.84e3·17-s + (−5.53e3 + 4.05e3i)19-s + 1.09e4i·21-s − 626.·23-s + 2.73e4·25-s + 5.60e4i·27-s + 2.21e4i·29-s + 5.36e3i·31-s + ⋯ |
L(s) = 1 | − 1.87i·3-s + 1.65·5-s − 0.631·7-s − 2.51·9-s − 0.953·11-s + 1.28i·13-s − 3.11i·15-s + 0.783·17-s + (−0.806 + 0.590i)19-s + 1.18i·21-s − 0.0515·23-s + 1.75·25-s + 2.84i·27-s + 0.908i·29-s + 0.180i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.806 - 0.590i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.806 - 0.590i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(1.175893391\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.175893391\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (5.53e3 - 4.05e3i)T \) |
good | 3 | \( 1 + 50.6iT - 729T^{2} \) |
| 5 | \( 1 - 207.T + 1.56e4T^{2} \) |
| 7 | \( 1 + 216.T + 1.17e5T^{2} \) |
| 11 | \( 1 + 1.26e3T + 1.77e6T^{2} \) |
| 13 | \( 1 - 2.82e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 - 3.84e3T + 2.41e7T^{2} \) |
| 23 | \( 1 + 626.T + 1.48e8T^{2} \) |
| 29 | \( 1 - 2.21e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 - 5.36e3iT - 8.87e8T^{2} \) |
| 37 | \( 1 - 4.07e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 - 2.60e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 1.34e5T + 6.32e9T^{2} \) |
| 47 | \( 1 + 1.56e5T + 1.07e10T^{2} \) |
| 53 | \( 1 + 2.43e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 - 3.61e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 4.34e5T + 5.15e10T^{2} \) |
| 67 | \( 1 + 2.17e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 - 1.25e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 1.30e5T + 1.51e11T^{2} \) |
| 79 | \( 1 - 4.31e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 - 6.18e5T + 3.26e11T^{2} \) |
| 89 | \( 1 - 1.89e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 5.89e5iT - 8.32e11T^{2} \) |
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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
−10.80426748320398197478620881917, −9.754339157884350655472399903712, −8.812614757331015657326330004657, −7.75826223713102299662498902074, −6.63694353300755342052254881744, −6.22042914323414075234718308390, −5.25931233366795358375797065289, −2.93950897182722808984990608544, −2.02157433341569991658126713275, −1.29359621438740290564581720435,
0.25899903523434920246191491594, 2.47369444316396222702557701029, 3.23484462375153947553772037393, 4.68039341752489820721306284573, 5.58840961963635036637444320031, 6.07867598087935275274435961721, 8.020260543966848470505978349427, 9.233465850223761823013588551434, 9.705093498595410109516877342355, 10.48173849465068433214372434881