| L(s) = 1 | − 8.33i·2-s + 16.8i·3-s − 5.39·4-s − 47.9i·5-s + 139.·6-s + 493. i·7-s − 488. i·8-s + 446.·9-s − 399.·10-s + 2.01e3·11-s − 90.6i·12-s − 124.·13-s + 4.11e3·14-s + 804.·15-s − 4.41e3·16-s + 3.85e3·17-s + ⋯ |
| L(s) = 1 | − 1.04i·2-s + 0.622i·3-s − 0.0843·4-s − 0.383i·5-s + 0.647·6-s + 1.43i·7-s − 0.953i·8-s + 0.612·9-s − 0.399·10-s + 1.51·11-s − 0.0524i·12-s − 0.0564·13-s + 1.49·14-s + 0.238·15-s − 1.07·16-s + 0.785·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.762 + 0.646i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.762 + 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.99206 - 0.730459i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.99206 - 0.730459i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (6.06e4 + 5.13e4i)T \) |
| good | 2 | \( 1 + 8.33iT - 64T^{2} \) |
| 3 | \( 1 - 16.8iT - 729T^{2} \) |
| 5 | \( 1 + 47.9iT - 1.56e4T^{2} \) |
| 7 | \( 1 - 493. iT - 1.17e5T^{2} \) |
| 11 | \( 1 - 2.01e3T + 1.77e6T^{2} \) |
| 13 | \( 1 + 124.T + 4.82e6T^{2} \) |
| 17 | \( 1 - 3.85e3T + 2.41e7T^{2} \) |
| 19 | \( 1 + 1.11e4iT - 4.70e7T^{2} \) |
| 23 | \( 1 + 1.13e3T + 1.48e8T^{2} \) |
| 29 | \( 1 - 2.43e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 + 1.24e4T + 8.87e8T^{2} \) |
| 37 | \( 1 - 8.46e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 - 1.25e5T + 4.75e9T^{2} \) |
| 47 | \( 1 + 2.00e5T + 1.07e10T^{2} \) |
| 53 | \( 1 + 1.24e5T + 2.21e10T^{2} \) |
| 59 | \( 1 + 1.23e5T + 4.21e10T^{2} \) |
| 61 | \( 1 + 2.87e4iT - 5.15e10T^{2} \) |
| 67 | \( 1 - 2.89e5T + 9.04e10T^{2} \) |
| 71 | \( 1 + 4.20e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 3.30e5iT - 1.51e11T^{2} \) |
| 79 | \( 1 + 6.47e5T + 2.43e11T^{2} \) |
| 83 | \( 1 + 7.67e4T + 3.26e11T^{2} \) |
| 89 | \( 1 + 1.69e4iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 1.10e6T + 8.32e11T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.73258861669851868094269376498, −12.88353478584526098059814595501, −12.06051062137669702178477943944, −11.11907996775533611028894900365, −9.639994611931490735103780988506, −8.973808739410430770920585706274, −6.66614791708295428360474149006, −4.78082878143752584815840741593, −3.16874342797266245680335170351, −1.42952992914123833648503340897,
1.36777951588150601660165496415, 4.01517503908801916029854601221, 6.19645310091569722254658749569, 7.10980877275988241921691790350, 7.909402194382690945283778232824, 9.880841114045650211163919849455, 11.26251484707128939586889924151, 12.65035655579655338221007404882, 14.21611385039064574166039860668, 14.48979949381818488432297805704
