L(s) = 1 | + (−5.54 − 1.12i)2-s + (15.2 − 3.36i)3-s + (29.4 + 12.4i)4-s + 25i·5-s + (−88.1 + 1.54i)6-s + 209. i·7-s + (−149. − 102. i)8-s + (220. − 102. i)9-s + (28.1 − 138. i)10-s − 618.·11-s + (490. + 90.6i)12-s − 94.2·13-s + (235. − 1.16e3i)14-s + (84.1 + 380. i)15-s + (712. + 735. i)16-s + 333. i·17-s + ⋯ |
L(s) = 1 | + (−0.980 − 0.198i)2-s + (0.976 − 0.216i)3-s + (0.920 + 0.389i)4-s + 0.447i·5-s + (−0.999 + 0.0175i)6-s + 1.61i·7-s + (−0.825 − 0.565i)8-s + (0.906 − 0.421i)9-s + (0.0889 − 0.438i)10-s − 1.54·11-s + (0.983 + 0.181i)12-s − 0.154·13-s + (0.321 − 1.58i)14-s + (0.0965 + 0.436i)15-s + (0.696 + 0.717i)16-s + 0.279i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.181 - 0.983i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.181 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.962204 + 0.800736i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.962204 + 0.800736i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (5.54 + 1.12i)T \) |
| 3 | \( 1 + (-15.2 + 3.36i)T \) |
| 5 | \( 1 - 25iT \) |
good | 7 | \( 1 - 209. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 618.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 94.2T + 3.71e5T^{2} \) |
| 17 | \( 1 - 333. iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 3.06e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 1.82e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 5.16e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 5.76e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 - 9.39e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 2.28e3iT - 1.15e8T^{2} \) |
| 43 | \( 1 + 2.57e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 1.62e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 8.56e3iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 1.78e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.26e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 8.19e3iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 4.90e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 9.21e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 5.42e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 - 7.22e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.04e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 + 1.17e4T + 8.58e9T^{2} \) |
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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
−14.73967533542410339523337198465, −12.94770422456442456187484928057, −12.12452541694436799127948167148, −10.59389535217902797454068441157, −9.517573616278195892977714500594, −8.409727348010728537691763283308, −7.60375602412427508256234610109, −5.90740048545864501654056864364, −3.05796248018923889351195393740, −2.05838376635618639751028116882,
0.71003977653287971493401622926, 2.72358978622529564637375441151, 4.74627325633966523772199425624, 7.12440638481459105412823865625, 7.83805547160848342520710360425, 9.088478096601186335974267199047, 10.18772029962033033480292458297, 10.99597653797346980804516576240, 13.03796013783066485067935221155, 13.82508442941284111612523674981