L(s) = 1 | + (−0.573 + 0.819i)5-s + (−0.819 + 0.573i)11-s + (0.819 + 0.573i)13-s + (−0.5 + 0.866i)17-s + (0.965 + 0.258i)19-s + (0.642 + 0.766i)23-s + (−0.342 − 0.939i)25-s + (−0.573 − 0.819i)29-s + (0.173 + 0.984i)31-s + (−0.707 − 0.707i)37-s + (−0.984 + 0.173i)41-s + (−0.0871 − 0.996i)43-s + (−0.173 + 0.984i)47-s + (0.258 − 0.965i)53-s − i·55-s + ⋯ |
L(s) = 1 | + (−0.573 + 0.819i)5-s + (−0.819 + 0.573i)11-s + (0.819 + 0.573i)13-s + (−0.5 + 0.866i)17-s + (0.965 + 0.258i)19-s + (0.642 + 0.766i)23-s + (−0.342 − 0.939i)25-s + (−0.573 − 0.819i)29-s + (0.173 + 0.984i)31-s + (−0.707 − 0.707i)37-s + (−0.984 + 0.173i)41-s + (−0.0871 − 0.996i)43-s + (−0.173 + 0.984i)47-s + (0.258 − 0.965i)53-s − i·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0322i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0322i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.01404846435 + 0.8721821327i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.01404846435 + 0.8721821327i\) |
\(L(1)\) |
\(\approx\) |
\(0.8042711376 + 0.3274056114i\) |
\(L(1)\) |
\(\approx\) |
\(0.8042711376 + 0.3274056114i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-0.573 + 0.819i)T \) |
| 11 | \( 1 + (-0.819 + 0.573i)T \) |
| 13 | \( 1 + (0.819 + 0.573i)T \) |
| 17 | \( 1 + (-0.5 + 0.866i)T \) |
| 19 | \( 1 + (0.965 + 0.258i)T \) |
| 23 | \( 1 + (0.642 + 0.766i)T \) |
| 29 | \( 1 + (-0.573 - 0.819i)T \) |
| 31 | \( 1 + (0.173 + 0.984i)T \) |
| 37 | \( 1 + (-0.707 - 0.707i)T \) |
| 41 | \( 1 + (-0.984 + 0.173i)T \) |
| 43 | \( 1 + (-0.0871 - 0.996i)T \) |
| 47 | \( 1 + (-0.173 + 0.984i)T \) |
| 53 | \( 1 + (0.258 - 0.965i)T \) |
| 59 | \( 1 + (0.422 + 0.906i)T \) |
| 61 | \( 1 + (0.573 + 0.819i)T \) |
| 67 | \( 1 + (-0.0871 + 0.996i)T \) |
| 71 | \( 1 + (-0.866 + 0.5i)T \) |
| 73 | \( 1 - iT \) |
| 79 | \( 1 + (0.766 + 0.642i)T \) |
| 83 | \( 1 + (0.573 + 0.819i)T \) |
| 89 | \( 1 + (-0.866 + 0.5i)T \) |
| 97 | \( 1 + (-0.766 - 0.642i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−17.323735625712187586801347287342, −16.44904285319498933165624509727, −16.211555110356575572441503176034, −15.42442563606743949864217677859, −15.046214735735768784272640916801, −13.81938513477683301609956837808, −13.42798639298468670155403134640, −12.869339630619628435949030810921, −12.099092628525014729299412960980, −11.36721535331709665268412525419, −10.937454210303596714803210246511, −10.07786419022664866943181088623, −9.21785154750955963086161473403, −8.66451501467529468273117344067, −8.055605129745913156268184177690, −7.42661933265977198327082900644, −6.613306263135924708942367971307, −5.670479062429666916277218816411, −5.06337636065832191254656242189, −4.5684639224265345525701852914, −3.390372815312425516008103350947, −3.12194477621293643886333772758, −1.944210646192295310966635479851, −0.94086106324171097477827569698, −0.261186833920449079490703072591,
1.22748469802573693485962825373, 2.0735105469410389499088368105, 2.88712100997744615030486470579, 3.69606895669038443099382465340, 4.15200337458162281366386189262, 5.22669407451060243115659790029, 5.80901009479592273180092591024, 6.88278595819304184305830961809, 7.09288739202975027066572912114, 8.01626774895206851150900290527, 8.5566318274184991499300718973, 9.452393616807710678013359200420, 10.24365442312384256456991797524, 10.72799454350591617192331005690, 11.46884312511198761346097663603, 11.93266583702278332726342461528, 12.85268803384472237413698451610, 13.46572200241004543743973735638, 14.11496889452404208892066158535, 14.87582866366491817574050333150, 15.49029932843683480520236582281, 15.85952462529152913310115163713, 16.63337877525316196355743939930, 17.65181879777584254576362145482, 17.93872757754801432656769963325