L(s) = 1 | + (0.342 − 0.939i)5-s + (−0.173 + 0.984i)7-s + (−0.342 − 0.939i)11-s + (−0.642 − 0.766i)13-s + (−0.5 − 0.866i)17-s + (0.866 + 0.5i)19-s + (−0.173 − 0.984i)23-s + (−0.766 − 0.642i)25-s + (0.642 − 0.766i)29-s + (0.173 + 0.984i)31-s + (0.866 + 0.5i)35-s + (0.866 − 0.5i)37-s + (−0.766 + 0.642i)41-s + (−0.342 − 0.939i)43-s + (0.173 − 0.984i)47-s + ⋯ |
L(s) = 1 | + (0.342 − 0.939i)5-s + (−0.173 + 0.984i)7-s + (−0.342 − 0.939i)11-s + (−0.642 − 0.766i)13-s + (−0.5 − 0.866i)17-s + (0.866 + 0.5i)19-s + (−0.173 − 0.984i)23-s + (−0.766 − 0.642i)25-s + (0.642 − 0.766i)29-s + (0.173 + 0.984i)31-s + (0.866 + 0.5i)35-s + (0.866 − 0.5i)37-s + (−0.766 + 0.642i)41-s + (−0.342 − 0.939i)43-s + (0.173 − 0.984i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0145 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0145 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7928101502 - 0.7813622437i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7928101502 - 0.7813622437i\) |
\(L(1)\) |
\(\approx\) |
\(0.9639202381 - 0.2854694852i\) |
\(L(1)\) |
\(\approx\) |
\(0.9639202381 - 0.2854694852i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (0.342 - 0.939i)T \) |
| 7 | \( 1 + (-0.173 + 0.984i)T \) |
| 11 | \( 1 + (-0.342 - 0.939i)T \) |
| 13 | \( 1 + (-0.642 - 0.766i)T \) |
| 17 | \( 1 + (-0.5 - 0.866i)T \) |
| 19 | \( 1 + (0.866 + 0.5i)T \) |
| 23 | \( 1 + (-0.173 - 0.984i)T \) |
| 29 | \( 1 + (0.642 - 0.766i)T \) |
| 31 | \( 1 + (0.173 + 0.984i)T \) |
| 37 | \( 1 + (0.866 - 0.5i)T \) |
| 41 | \( 1 + (-0.766 + 0.642i)T \) |
| 43 | \( 1 + (-0.342 - 0.939i)T \) |
| 47 | \( 1 + (0.173 - 0.984i)T \) |
| 53 | \( 1 + iT \) |
| 59 | \( 1 + (0.342 - 0.939i)T \) |
| 61 | \( 1 + (-0.984 - 0.173i)T \) |
| 67 | \( 1 + (-0.642 - 0.766i)T \) |
| 71 | \( 1 + (0.5 + 0.866i)T \) |
| 73 | \( 1 + (0.5 - 0.866i)T \) |
| 79 | \( 1 + (0.766 + 0.642i)T \) |
| 83 | \( 1 + (0.642 - 0.766i)T \) |
| 89 | \( 1 + (0.5 - 0.866i)T \) |
| 97 | \( 1 + (-0.939 + 0.342i)T \) |
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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
−24.10683441294255760871892443646, −23.528753211680323373399741714329, −22.528969167971613732551080699432, −21.92898731211695913379319143156, −20.94065601977845662074502789837, −19.89131666129287618403563950446, −19.32312506666557153070772398919, −18.10234759633002863835784109864, −17.53760227170348783524493095338, −16.64982394183802270364053322698, −15.468329568079290237698282266434, −14.70239136645622427936637288270, −13.79665873562066822242348097305, −13.09950241492459386791724430007, −11.83216287398582761591928139208, −10.91488783200345551377188685747, −10.02603417338391284243043828809, −9.43073829112986773463155564815, −7.779564815111475147305458372782, −7.09936818036079274931431558537, −6.31994131054863848716391463553, −4.91542461498894830938861963619, −3.90653711323779092140406769996, −2.73692125842586902220691657740, −1.60314581419302753041993325939,
0.63181574499973800375185392659, 2.227001960423105528575070793641, 3.1729408592196611458988969010, 4.79134070924590692776310373968, 5.44944206845933346751457437489, 6.36300283056078657889373765344, 7.87076457626185810138749568890, 8.63312670272467869584300378350, 9.46761029320035384054526297819, 10.410345257314010896241887793016, 11.767714407040001052251164089126, 12.35554699634398743061311334485, 13.319552830929981421190315891497, 14.12545803059003783516976739148, 15.389029055549409327327337815368, 16.073510662696040334230571367335, 16.84861884942329719110404363650, 17.98578532160650166873538817708, 18.574368753641461638240472961748, 19.759015522826036082769863244656, 20.450289361821588245026941910824, 21.429889934792033946950606140437, 22.042340015984249494338412135267, 23.008765957566311393833169829296, 24.171472485853383393429296068583