| L(s) = 1 | + 42.5i·3-s + 103.·5-s + 619.·7-s − 1.08e3·9-s + 57.6·11-s − 481. i·13-s + 4.40e3i·15-s − 2.33e3·17-s + (−4.35e3 − 5.29e3i)19-s + 2.63e4i·21-s + 8.69e3·23-s − 4.92e3·25-s − 1.51e4i·27-s + 4.60e4i·29-s + 3.67e4i·31-s + ⋯ |
| L(s) = 1 | + 1.57i·3-s + 0.827·5-s + 1.80·7-s − 1.48·9-s + 0.0432·11-s − 0.219i·13-s + 1.30i·15-s − 0.476·17-s + (−0.635 − 0.772i)19-s + 2.84i·21-s + 0.714·23-s − 0.315·25-s − 0.771i·27-s + 1.88i·29-s + 1.23i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.635 - 0.772i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.635 - 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.930464510\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.930464510\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (4.35e3 + 5.29e3i)T \) |
| good | 3 | \( 1 - 42.5iT - 729T^{2} \) |
| 5 | \( 1 - 103.T + 1.56e4T^{2} \) |
| 7 | \( 1 - 619.T + 1.17e5T^{2} \) |
| 11 | \( 1 - 57.6T + 1.77e6T^{2} \) |
| 13 | \( 1 + 481. iT - 4.82e6T^{2} \) |
| 17 | \( 1 + 2.33e3T + 2.41e7T^{2} \) |
| 23 | \( 1 - 8.69e3T + 1.48e8T^{2} \) |
| 29 | \( 1 - 4.60e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 - 3.67e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 + 4.04e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 - 6.15e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 6.43e4T + 6.32e9T^{2} \) |
| 47 | \( 1 - 9.39e4T + 1.07e10T^{2} \) |
| 53 | \( 1 - 9.78e4iT - 2.21e10T^{2} \) |
| 59 | \( 1 - 1.32e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 1.92e3T + 5.15e10T^{2} \) |
| 67 | \( 1 - 3.78e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 - 4.51e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 6.69e5T + 1.51e11T^{2} \) |
| 79 | \( 1 - 7.47e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 - 4.95e5T + 3.26e11T^{2} \) |
| 89 | \( 1 + 1.35e6iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 7.59e4iT - 8.32e11T^{2} \) |
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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
−10.82093290930580351412532839736, −10.30200338945549260106575935878, −9.002240404872516544309185708774, −8.697840641109999592173964655283, −7.21820655402020296458167988782, −5.62618945420839100414557439631, −4.94404979907874688557759556572, −4.20636958859454677911494479369, −2.70020522590482620596653464303, −1.38363723556910175189505193106,
0.69617938208481102212027228060, 1.86401817690244553997929275535, 2.19048956725235871012191104231, 4.37428691228104870052847376139, 5.64752103435590637037120370469, 6.41093790963073015018813990896, 7.63556645736938837821375005374, 8.101319307298172555123646720778, 9.176549573692166672338166717863, 10.56484710482504261850369674562
