| L(s) = 1 | + (−0.853 + 0.853i)2-s + (11.3 + 11.3i)3-s + 62.5i·4-s − 19.3·6-s + (454. − 454. i)7-s + (−107. − 107. i)8-s − 471. i·9-s − 1.60e3·11-s + (−709. + 709. i)12-s + (−1.32e3 − 1.32e3i)13-s + 774. i·14-s − 3.81e3·16-s + (−4.41e3 + 4.41e3i)17-s + (402. + 402. i)18-s − 4.18e3i·19-s + ⋯ |
| L(s) = 1 | + (−0.106 + 0.106i)2-s + (0.420 + 0.420i)3-s + 0.977i·4-s − 0.0895·6-s + (1.32 − 1.32i)7-s + (−0.210 − 0.210i)8-s − 0.647i·9-s − 1.20·11-s + (−0.410 + 0.410i)12-s + (−0.601 − 0.601i)13-s + 0.282i·14-s − 0.932·16-s + (−0.899 + 0.899i)17-s + (0.0690 + 0.0690i)18-s − 0.610i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.146482446\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.146482446\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 + (0.853 - 0.853i)T - 64iT^{2} \) |
| 3 | \( 1 + (-11.3 - 11.3i)T + 729iT^{2} \) |
| 7 | \( 1 + (-454. + 454. i)T - 1.17e5iT^{2} \) |
| 11 | \( 1 + 1.60e3T + 1.77e6T^{2} \) |
| 13 | \( 1 + (1.32e3 + 1.32e3i)T + 4.82e6iT^{2} \) |
| 17 | \( 1 + (4.41e3 - 4.41e3i)T - 2.41e7iT^{2} \) |
| 19 | \( 1 + 4.18e3iT - 4.70e7T^{2} \) |
| 23 | \( 1 + (1.24e4 + 1.24e4i)T + 1.48e8iT^{2} \) |
| 29 | \( 1 + 3.76e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 - 6.76e3T + 8.87e8T^{2} \) |
| 37 | \( 1 + (6.21e4 - 6.21e4i)T - 2.56e9iT^{2} \) |
| 41 | \( 1 + 6.74e4T + 4.75e9T^{2} \) |
| 43 | \( 1 + (-6.35e4 - 6.35e4i)T + 6.32e9iT^{2} \) |
| 47 | \( 1 + (-2.37e4 + 2.37e4i)T - 1.07e10iT^{2} \) |
| 53 | \( 1 + (-1.46e4 - 1.46e4i)T + 2.21e10iT^{2} \) |
| 59 | \( 1 - 1.19e4iT - 4.21e10T^{2} \) |
| 61 | \( 1 - 1.43e5T + 5.15e10T^{2} \) |
| 67 | \( 1 + (-1.01e5 + 1.01e5i)T - 9.04e10iT^{2} \) |
| 71 | \( 1 - 3.25e5T + 1.28e11T^{2} \) |
| 73 | \( 1 + (-4.39e5 - 4.39e5i)T + 1.51e11iT^{2} \) |
| 79 | \( 1 + 2.53e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + (1.66e5 + 1.66e5i)T + 3.26e11iT^{2} \) |
| 89 | \( 1 - 5.88e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + (1.47e5 - 1.47e5i)T - 8.32e11iT^{2} \) |
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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
−11.98781676114113572181407559028, −10.83612537432403899254741822024, −9.986250174519203689412661571396, −8.352691579656043943050618853049, −8.001500992916640648759436005441, −6.76983432098524096594318940697, −4.73160146427819893099284749003, −3.91029261793503574368089813201, −2.43734633401393260805357395561, −0.32591998725803744630557105450,
1.82756109846215847473401709356, 2.33028256198674898282627682266, 4.98529930862948510560686197666, 5.45948740407641017176174137724, 7.23388823463691751750554479914, 8.326898196506611987115972874238, 9.200571141967591723540786541937, 10.53314272329715024603665938456, 11.39301317955431293830620516720, 12.40080592756469805995024117874
