L(s) = 1 | − 17.7i·2-s + 152. i·3-s − 59.7·4-s − 783. i·5-s + 2.70e3·6-s + 1.14e3i·7-s − 3.48e3i·8-s − 1.66e4·9-s − 1.39e4·10-s + 2.13e4·11-s − 9.11e3i·12-s + 1.69e4·13-s + 2.04e4·14-s + 1.19e5·15-s − 7.72e4·16-s + 1.04e5·17-s + ⋯ |
L(s) = 1 | − 1.11i·2-s + 1.88i·3-s − 0.233·4-s − 1.25i·5-s + 2.09·6-s + 0.478i·7-s − 0.851i·8-s − 2.54·9-s − 1.39·10-s + 1.45·11-s − 0.439i·12-s + 0.593·13-s + 0.531·14-s + 2.35·15-s − 1.17·16-s + 1.25·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 + 0.357i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.934 + 0.357i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(2.15021 - 0.396953i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.15021 - 0.396953i\) |
\(L(5)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 43 | \( 1 + (-3.19e6 - 1.22e6i)T \) |
good | 2 | \( 1 + 17.7iT - 256T^{2} \) |
| 3 | \( 1 - 152. iT - 6.56e3T^{2} \) |
| 5 | \( 1 + 783. iT - 3.90e5T^{2} \) |
| 7 | \( 1 - 1.14e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 - 2.13e4T + 2.14e8T^{2} \) |
| 13 | \( 1 - 1.69e4T + 8.15e8T^{2} \) |
| 17 | \( 1 - 1.04e5T + 6.97e9T^{2} \) |
| 19 | \( 1 - 1.42e5iT - 1.69e10T^{2} \) |
| 23 | \( 1 - 3.76e5T + 7.83e10T^{2} \) |
| 29 | \( 1 - 2.45e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 - 3.73e5T + 8.52e11T^{2} \) |
| 37 | \( 1 + 1.72e6iT - 3.51e12T^{2} \) |
| 41 | \( 1 + 3.57e6T + 7.98e12T^{2} \) |
| 47 | \( 1 - 6.89e6T + 2.38e13T^{2} \) |
| 53 | \( 1 + 3.52e6T + 6.22e13T^{2} \) |
| 59 | \( 1 + 1.55e6T + 1.46e14T^{2} \) |
| 61 | \( 1 + 3.86e6iT - 1.91e14T^{2} \) |
| 67 | \( 1 + 1.77e7T + 4.06e14T^{2} \) |
| 71 | \( 1 + 3.92e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 2.65e7iT - 8.06e14T^{2} \) |
| 79 | \( 1 + 6.70e7T + 1.51e15T^{2} \) |
| 83 | \( 1 + 1.73e7T + 2.25e15T^{2} \) |
| 89 | \( 1 + 1.06e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + 1.23e8T + 7.83e15T^{2} \) |
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
−14.25431384180351536379978983920, −12.43102473160243337103342257092, −11.64235381387112878653906933526, −10.48326216163561258176849596688, −9.387115976116497618515946324363, −8.797110379706876942286827767675, −5.68328081834471658282372555570, −4.29637092651803126262031282345, −3.34385661622044802702724956537, −1.14107291358834451033293268753,
1.16312553609002665318239927140, 2.88201175902975034400509664094, 5.96113233723408317959550235476, 6.90020924218322978542255412606, 7.26417471491633468303183282689, 8.652583545725139253680899484418, 11.05928643865594972105803760425, 11.93311426541849231734685218000, 13.57842939822972120197919504349, 14.22577280516725884858979106423