L(s) = 1 | − 19.2i·2-s + (46.1 + 7.40i)3-s − 242.·4-s + 153.·5-s + (142. − 888. i)6-s − 1.44e3i·7-s + 2.20e3i·8-s + (2.07e3 + 683. i)9-s − 2.95e3i·10-s + 5.06e3·11-s + (−1.11e4 − 1.79e3i)12-s + 9.56e3·13-s − 2.77e4·14-s + (7.10e3 + 1.13e3i)15-s + 1.13e4·16-s − 2.37e4·17-s + ⋯ |
L(s) = 1 | − 1.70i·2-s + (0.987 + 0.158i)3-s − 1.89·4-s + 0.550·5-s + (0.269 − 1.67i)6-s − 1.59i·7-s + 1.52i·8-s + (0.949 + 0.312i)9-s − 0.935i·10-s + 1.14·11-s + (−1.86 − 0.299i)12-s + 1.20·13-s − 2.70·14-s + (0.543 + 0.0871i)15-s + 0.692·16-s − 1.17·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0822i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.996 + 0.0822i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(0.114086 - 2.76959i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.114086 - 2.76959i\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-46.1 - 7.40i)T \) |
| 23 | \( 1 + (5.66e4 - 1.39e4i)T \) |
good | 2 | \( 1 + 19.2iT - 128T^{2} \) |
| 5 | \( 1 - 153.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.44e3iT - 8.23e5T^{2} \) |
| 11 | \( 1 - 5.06e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 9.56e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.37e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 3.39e4iT - 8.93e8T^{2} \) |
| 29 | \( 1 - 1.13e5iT - 1.72e10T^{2} \) |
| 31 | \( 1 + 3.66e3T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.86e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 + 4.03e5iT - 1.94e11T^{2} \) |
| 43 | \( 1 - 2.63e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 - 1.77e5iT - 5.06e11T^{2} \) |
| 53 | \( 1 - 1.86e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.81e5iT - 2.48e12T^{2} \) |
| 61 | \( 1 + 2.62e6iT - 3.14e12T^{2} \) |
| 67 | \( 1 - 2.20e6iT - 6.06e12T^{2} \) |
| 71 | \( 1 - 5.42e6iT - 9.09e12T^{2} \) |
| 73 | \( 1 - 5.37e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 4.32e6iT - 1.92e13T^{2} \) |
| 83 | \( 1 + 2.97e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 2.86e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.05e6iT - 8.07e13T^{2} \) |
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
−12.97716268868820980765683553086, −11.33627660692831945920187467973, −10.53227846446519496258580227269, −9.546025256475456766804193771193, −8.688678536115286647542576176483, −6.89984742441093592524713224824, −4.25546264905588459971435966954, −3.64897359209222710706709425682, −2.01142439721605592705413358875, −0.935919064777645829604242118036,
1.97589826643682789596611645419, 4.05150290621989732271114915436, 5.87075824210108715668679193057, 6.49708862758631182149630632418, 8.171645266601036879113641478822, 8.791370458701207476140056768195, 9.625246512256143867687615002677, 11.95689919454562841428030627377, 13.36721277845264913976636990094, 14.06339817513796104823578728122