L(s) = 1 | + (0.866 − 0.5i)2-s + (0.130 − 0.991i)3-s + (0.499 − 0.866i)4-s + (−0.382 − 0.923i)6-s − 0.999i·8-s + (−0.965 − 0.258i)9-s + (1.22 + 0.707i)11-s + (−0.793 − 0.608i)12-s + (−0.5 − 0.866i)16-s + (0.382 − 0.662i)17-s + (−0.965 + 0.258i)18-s + (−1.60 + 0.923i)19-s + 1.41·22-s + (−0.991 − 0.130i)24-s + (−0.5 + 0.866i)25-s + ⋯ |
L(s) = 1 | + (0.866 − 0.5i)2-s + (0.130 − 0.991i)3-s + (0.499 − 0.866i)4-s + (−0.382 − 0.923i)6-s − 0.999i·8-s + (−0.965 − 0.258i)9-s + (1.22 + 0.707i)11-s + (−0.793 − 0.608i)12-s + (−0.5 − 0.866i)16-s + (0.382 − 0.662i)17-s + (−0.965 + 0.258i)18-s + (−1.60 + 0.923i)19-s + 1.41·22-s + (−0.991 − 0.130i)24-s + (−0.5 + 0.866i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.294 + 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.294 + 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.769766599\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.769766599\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 + 0.5i)T \) |
| 3 | \( 1 + (-0.130 + 0.991i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + (-0.382 + 0.662i)T + (-0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (1.60 - 0.923i)T + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 - 1.84T + T^{2} \) |
| 43 | \( 1 + 1.41T + T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.382 - 0.662i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (-0.662 - 0.382i)T + (0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - 0.765T + T^{2} \) |
| 89 | \( 1 + (-0.923 - 1.60i)T + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + 1.84iT - T^{2} \) |
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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
−9.744202846385355430366826334366, −9.047003829998635012710287666176, −7.899619750505257566538310238750, −7.00430900569692073246960798974, −6.36706563556503414648159052194, −5.58343019021888808855797014833, −4.38321072744372134252689648740, −3.50710210725827676929358906334, −2.27683047824529867166977844547, −1.40469594582521019491252182490,
2.32379401628011227533442206774, 3.51865376714167412029723928370, 4.14665276207051570727330971917, 4.94424778159783348198474763519, 6.12905902649710301633953981380, 6.43600140648504717068467578711, 7.83348092127495797839870548432, 8.605394504170754823074743508465, 9.176872214931083523221858111914, 10.35261714350855680500324308637