L(s) = 1 | + (0.965 + 0.258i)2-s + (0.382 + 0.923i)3-s + (0.866 + 0.499i)4-s + (−0.608 + 0.793i)5-s + (0.130 + 0.991i)6-s + (0.258 − 0.965i)7-s + (0.707 + 0.707i)8-s + (−0.707 + 0.707i)9-s + (−0.793 + 0.608i)10-s + (−0.130 + 0.991i)12-s + (0.478 + 1.78i)13-s + (0.499 − 0.866i)14-s + (−0.965 − 0.258i)15-s + (0.500 + 0.866i)16-s + (−0.866 + 0.500i)18-s − 1.98i·19-s + ⋯ |
L(s) = 1 | + (0.965 + 0.258i)2-s + (0.382 + 0.923i)3-s + (0.866 + 0.499i)4-s + (−0.608 + 0.793i)5-s + (0.130 + 0.991i)6-s + (0.258 − 0.965i)7-s + (0.707 + 0.707i)8-s + (−0.707 + 0.707i)9-s + (−0.793 + 0.608i)10-s + (−0.130 + 0.991i)12-s + (0.478 + 1.78i)13-s + (0.499 − 0.866i)14-s + (−0.965 − 0.258i)15-s + (0.500 + 0.866i)16-s + (−0.866 + 0.500i)18-s − 1.98i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.329 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.329 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.294736650\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.294736650\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.965 - 0.258i)T \) |
| 3 | \( 1 + (-0.382 - 0.923i)T \) |
| 5 | \( 1 + (0.608 - 0.793i)T \) |
| 7 | \( 1 + (-0.258 + 0.965i)T \) |
good | 11 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.478 - 1.78i)T + (-0.866 + 0.5i)T^{2} \) |
| 17 | \( 1 + iT^{2} \) |
| 19 | \( 1 + 1.98iT - T^{2} \) |
| 23 | \( 1 + (0.5 - 0.133i)T + (0.866 - 0.5i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 47 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 + (-0.382 + 0.662i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.130 + 0.226i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 71 | \( 1 - 1.93iT - T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 + (-1.67 + 0.965i)T + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.198 + 0.739i)T + (-0.866 - 0.5i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (0.866 + 0.5i)T^{2} \) |
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
−9.291527230331899782300672553909, −8.458460231275941374056466004252, −7.62023436269209697774136373697, −6.91630525770725912727188288363, −6.40092366266103758859356363048, −5.08999307742317589158422232656, −4.33131583782256054828781863373, −3.95549805909565259893832719553, −3.07176535142477312355869450923, −2.09358615878723980855332713572,
1.14084151304895837938569060131, 2.12076977558451136511034873231, 3.20166978317967047407654635622, 3.82121477555478596088597636190, 5.08573742853197087309469141597, 5.75214878680225419748373146945, 6.20802486231308867843994567634, 7.52825035058124391056733211823, 8.028918364606835856315513329437, 8.520301342619341803600030484163