L(s) = 1 | + 15.4·2-s + 173.·4-s − 161. i·5-s + 585. i·7-s + 1.68e3·8-s − 2.48e3i·10-s − 2.12e3i·11-s + 1.92e3·13-s + 9.02e3i·14-s + 1.48e4·16-s + 1.37e3i·17-s − 3.65e3i·19-s − 2.79e4i·20-s − 3.26e4i·22-s + (1.09e4 + 5.40e3i)23-s + ⋯ |
L(s) = 1 | + 1.92·2-s + 2.70·4-s − 1.29i·5-s + 1.70i·7-s + 3.29·8-s − 2.48i·10-s − 1.59i·11-s + 0.877·13-s + 3.28i·14-s + 3.62·16-s + 0.279i·17-s − 0.532i·19-s − 3.49i·20-s − 3.06i·22-s + (0.895 + 0.444i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(8.004416117\) |
\(L(\frac12)\) |
\(\approx\) |
\(8.004416117\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 23 | \( 1 + (-1.09e4 - 5.40e3i)T \) |
good | 2 | \( 1 - 15.4T + 64T^{2} \) |
| 5 | \( 1 + 161. iT - 1.56e4T^{2} \) |
| 7 | \( 1 - 585. iT - 1.17e5T^{2} \) |
| 11 | \( 1 + 2.12e3iT - 1.77e6T^{2} \) |
| 13 | \( 1 - 1.92e3T + 4.82e6T^{2} \) |
| 17 | \( 1 - 1.37e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 3.65e3iT - 4.70e7T^{2} \) |
| 29 | \( 1 + 6.03e3T + 5.94e8T^{2} \) |
| 31 | \( 1 - 3.25e3T + 8.87e8T^{2} \) |
| 37 | \( 1 - 2.55e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + 1.01e5T + 4.75e9T^{2} \) |
| 43 | \( 1 + 8.34e4iT - 6.32e9T^{2} \) |
| 47 | \( 1 - 1.19e5T + 1.07e10T^{2} \) |
| 53 | \( 1 + 8.70e4iT - 2.21e10T^{2} \) |
| 59 | \( 1 + 4.21e4T + 4.21e10T^{2} \) |
| 61 | \( 1 - 1.12e5iT - 5.15e10T^{2} \) |
| 67 | \( 1 - 4.98e4iT - 9.04e10T^{2} \) |
| 71 | \( 1 + 2.34e5T + 1.28e11T^{2} \) |
| 73 | \( 1 + 7.52e4T + 1.51e11T^{2} \) |
| 79 | \( 1 - 5.39e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 - 2.21e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 - 1.34e6iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 1.22e6iT - 8.32e11T^{2} \) |
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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
−11.71007443165486792954176779961, −10.91801761937211619269887466871, −8.970257800917671467676008848274, −8.304732135028462520440421865878, −6.50027014749811320887320457131, −5.53870007938834481877949040027, −5.19619722034779010396487099123, −3.74413374651567392146950308636, −2.68568423352350334486210935442, −1.29394646382771342217187984356,
1.60021611150044728161965996734, 2.98922380500229737507464298882, 3.91077372332492714875498419344, 4.72501491431918154766570448787, 6.26137632395996979213597152264, 7.07004029646504530710770106603, 7.48278995224925069930671730758, 10.16020530402347956104543413734, 10.67973243802045436061587731223, 11.46415937044973700755170607561