L(s) = 1 | + (0.866 − 0.5i)4-s + (0.866 + 0.5i)9-s + (−1 − 1.73i)11-s + (0.499 − 0.866i)16-s + 2i·29-s + 0.999·36-s + (−1.73 − 0.999i)44-s − 0.999i·64-s − 2·71-s + (1.73 + i)79-s + (0.499 + 0.866i)81-s − 1.99i·99-s + (−1.73 + i)109-s + (1 + 1.73i)116-s + ⋯ |
L(s) = 1 | + (0.866 − 0.5i)4-s + (0.866 + 0.5i)9-s + (−1 − 1.73i)11-s + (0.499 − 0.866i)16-s + 2i·29-s + 0.999·36-s + (−1.73 − 0.999i)44-s − 0.999i·64-s − 2·71-s + (1.73 + i)79-s + (0.499 + 0.866i)81-s − 1.99i·99-s + (−1.73 + i)109-s + (1 + 1.73i)116-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.813 + 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.813 + 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.297508695\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.297508695\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 3 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 11 | \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 29 | \( 1 - 2iT - T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 53 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 71 | \( 1 + 2T + T^{2} \) |
| 73 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 79 | \( 1 + (-1.73 - i)T + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 - iT^{2} \) |
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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
−10.10842161674811469590070569971, −9.010575697299198735540361607344, −8.102313856268269707268245424081, −7.36073443779792664314877648857, −6.52911788623658706797619758861, −5.61815753668855137836644432381, −4.97329414844732307864484567044, −3.49584534378517935283434112741, −2.58310175237063391028245005097, −1.29453476774021505052523383517,
1.80357785514661503823830905434, 2.66488087310503864355433301927, 3.94965068992227505423442553231, 4.73532175608862269840415202864, 5.98991009960892986026453138808, 6.88038970613669655230438498838, 7.50046853287663568076180233908, 8.070986283820880948501133195982, 9.373894503014709617359903289575, 10.06624578909594598452719613275