L(s) = 1 | − 1.41i·5-s + 2i·7-s + 1.41·11-s + 1.41i·17-s + i·19-s − 1.41·23-s − 1.00·25-s + 2.82·35-s + 1.41·47-s − 3·49-s − 2.00i·55-s + 2·73-s + 2.82i·77-s + 1.41·83-s + 2.00·85-s + ⋯ |
L(s) = 1 | − 1.41i·5-s + 2i·7-s + 1.41·11-s + 1.41i·17-s + i·19-s − 1.41·23-s − 1.00·25-s + 2.82·35-s + 1.41·47-s − 3·49-s − 2.00i·55-s + 2·73-s + 2.82i·77-s + 1.41·83-s + 2.00·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.259264460\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.259264460\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 - iT \) |
good | 5 | \( 1 + 1.41iT - T^{2} \) |
| 7 | \( 1 - 2iT - T^{2} \) |
| 11 | \( 1 - 1.41T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - 1.41iT - T^{2} \) |
| 23 | \( 1 + 1.41T + T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - 1.41T + T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - 2T + T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 - 1.41T + T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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
−9.136361087223426000460471194020, −8.326642673020094850144310935679, −8.070189289378602296060400796476, −6.46984049397799711229255896215, −5.90041848213770564794817294368, −5.39359447661659859675260841311, −4.33874956176588278095412910090, −3.63931842089723274850509806091, −2.14323970983564923887728631969, −1.49038406039032023729041155104,
0.883878063637546564952788224397, 2.34550295344541057778749810199, 3.46770985732559170356566793295, 3.96457280499770947401462880832, 4.83201538347511330668355389482, 6.26070675739910094445577269080, 6.78419744042558602427863705789, 7.25760781421313360805833510494, 7.84731530472322520093311834301, 9.132873809798500117072447077502