L(s) = 1 | + (−0.342 + 0.939i)2-s + (−0.766 − 0.642i)4-s + (−0.866 − 0.5i)7-s + (0.866 − 0.5i)8-s + (0.5 + 0.866i)11-s + (−0.984 − 0.173i)13-s + (0.766 − 0.642i)14-s + (0.173 + 0.984i)16-s + (−0.342 + 0.939i)17-s + (−0.984 + 0.173i)22-s + (−0.642 + 0.766i)23-s + (0.5 − 0.866i)26-s + (0.342 + 0.939i)28-s + (−0.939 + 0.342i)29-s + (−0.5 + 0.866i)31-s + (−0.984 − 0.173i)32-s + ⋯ |
L(s) = 1 | + (−0.342 + 0.939i)2-s + (−0.766 − 0.642i)4-s + (−0.866 − 0.5i)7-s + (0.866 − 0.5i)8-s + (0.5 + 0.866i)11-s + (−0.984 − 0.173i)13-s + (0.766 − 0.642i)14-s + (0.173 + 0.984i)16-s + (−0.342 + 0.939i)17-s + (−0.984 + 0.173i)22-s + (−0.642 + 0.766i)23-s + (0.5 − 0.866i)26-s + (0.342 + 0.939i)28-s + (−0.939 + 0.342i)29-s + (−0.5 + 0.866i)31-s + (−0.984 − 0.173i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0141i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0141i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.002740097723 + 0.3867409698i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.002740097723 + 0.3867409698i\) |
\(L(1)\) |
\(\approx\) |
\(0.5195746906 + 0.3144498518i\) |
\(L(1)\) |
\(\approx\) |
\(0.5195746906 + 0.3144498518i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (-0.342 + 0.939i)T \) |
| 7 | \( 1 + (-0.866 - 0.5i)T \) |
| 11 | \( 1 + (0.5 + 0.866i)T \) |
| 13 | \( 1 + (-0.984 - 0.173i)T \) |
| 17 | \( 1 + (-0.342 + 0.939i)T \) |
| 23 | \( 1 + (-0.642 + 0.766i)T \) |
| 29 | \( 1 + (-0.939 + 0.342i)T \) |
| 31 | \( 1 + (-0.5 + 0.866i)T \) |
| 37 | \( 1 + iT \) |
| 41 | \( 1 + (-0.173 - 0.984i)T \) |
| 43 | \( 1 + (-0.642 - 0.766i)T \) |
| 47 | \( 1 + (0.342 + 0.939i)T \) |
| 53 | \( 1 + (-0.642 + 0.766i)T \) |
| 59 | \( 1 + (-0.939 - 0.342i)T \) |
| 61 | \( 1 + (0.766 + 0.642i)T \) |
| 67 | \( 1 + (-0.342 - 0.939i)T \) |
| 71 | \( 1 + (-0.766 + 0.642i)T \) |
| 73 | \( 1 + (0.984 - 0.173i)T \) |
| 79 | \( 1 + (-0.173 - 0.984i)T \) |
| 83 | \( 1 + (-0.866 - 0.5i)T \) |
| 89 | \( 1 + (0.173 - 0.984i)T \) |
| 97 | \( 1 + (0.342 - 0.939i)T \) |
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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
−25.1637048189712244855222329312, −24.32096028675099132502592834664, −22.88646263982842287643947098741, −22.17563597377543911334563309156, −21.619369509745839178188605765031, −20.38775129145487120264115931773, −19.62422175908373318159043948942, −18.85013935500457135979728125555, −18.1017639422802914755928255391, −16.825400156690305608776563466220, −16.2525863800362967193718657798, −14.77302241596856445746886655016, −13.70729626413754538529417067813, −12.79905079662677702134904489579, −11.898243294499381715430298249632, −11.11442268627249974028433766271, −9.78278960722947876060631012875, −9.28365515123617623885440903646, −8.17334428991079969502326161735, −6.89025805335273880356957004222, −5.557624885067614501540371093311, −4.21823881192530741114830668613, −3.08860387376699605931930217675, −2.133242723016792786771468190760, −0.282640639302088837589347570116,
1.68695593320861209744542635032, 3.616016387707683546526450096286, 4.68872115546161530778325127366, 5.92739715468942214084799129601, 6.93911448659228000098267769276, 7.61112557134239443108242623622, 8.97491468797988669441274586408, 9.80021324960550192141470327267, 10.56812774121267364974627945955, 12.25526098461698402612442617005, 13.12795017884380419381109734505, 14.18698673157070631961648588265, 15.08064699739062261986618126341, 15.88527703759745093056683940299, 17.060109533461666254065091093108, 17.3724456625091181578139849468, 18.65045230924070140655955044988, 19.62324589910641184840208418618, 20.15342293763942264456823053580, 21.97490623300827132821582427653, 22.44752713489861855810946440249, 23.516723608320519271859618922480, 24.17690709361917764352207523504, 25.364364766175300733869577125929, 25.814527546073727400421424421778