L(s) = 1 | − 4-s + 1.73i·13-s + 16-s + 25-s + 1.73i·31-s + 37-s + 43-s − 1.73i·52-s + 1.73i·61-s − 64-s − 67-s − 79-s − 1.73i·97-s − 100-s − 1.73i·103-s + ⋯ |
L(s) = 1 | − 4-s + 1.73i·13-s + 16-s + 25-s + 1.73i·31-s + 37-s + 43-s − 1.73i·52-s + 1.73i·61-s − 64-s − 67-s − 79-s − 1.73i·97-s − 100-s − 1.73i·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.654 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.654 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8183141478\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8183141478\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 - 1.73iT - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 - 1.73iT - T^{2} \) |
| 37 | \( 1 - T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - 1.73iT - T^{2} \) |
| 67 | \( 1 + T + T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + T + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + 1.73iT - T^{2} \) |
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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
−9.810582568381116554670778374717, −8.952418652468414137339544634760, −8.685647739212262385571700418527, −7.48797561481281232731281768130, −6.70125587424120817066982268264, −5.70013012337693877383196838221, −4.67077253986086820766895656586, −4.14792620377364889965110539767, −2.93666563710702309308244415973, −1.41297604436463023655939727867,
0.816458466599445472109253402415, 2.67887786548312675246589072208, 3.67752577471363120000324818609, 4.65526940162485078564293803722, 5.47557457589769671937395229666, 6.21431225337471449681437970363, 7.60297882088044127186317709461, 8.045690170380068068143468398716, 8.956214338898187121593305380183, 9.673430695217100229039688076918