L(s) = 1 | − 2-s + 4-s − 8-s + (−0.5 − 0.866i)11-s + (0.5 + 0.866i)13-s + 16-s + (0.5 − 0.866i)17-s + (−0.5 − 0.866i)19-s + (0.5 + 0.866i)22-s + (0.5 − 0.866i)23-s + (−0.5 − 0.866i)26-s + (−0.5 + 0.866i)29-s + 31-s − 32-s + (−0.5 + 0.866i)34-s + ⋯ |
L(s) = 1 | − 2-s + 4-s − 8-s + (−0.5 − 0.866i)11-s + (0.5 + 0.866i)13-s + 16-s + (0.5 − 0.866i)17-s + (−0.5 − 0.866i)19-s + (0.5 + 0.866i)22-s + (0.5 − 0.866i)23-s + (−0.5 − 0.866i)26-s + (−0.5 + 0.866i)29-s + 31-s − 32-s + (−0.5 + 0.866i)34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.690 - 0.723i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.690 - 0.723i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6973718801 - 0.2986697347i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6973718801 - 0.2986697347i\) |
\(L(1)\) |
\(\approx\) |
\(0.7029183920 - 0.09803029914i\) |
\(L(1)\) |
\(\approx\) |
\(0.7029183920 - 0.09803029914i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - T \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
| 13 | \( 1 + (0.5 + 0.866i)T \) |
| 17 | \( 1 + (0.5 - 0.866i)T \) |
| 19 | \( 1 + (-0.5 - 0.866i)T \) |
| 23 | \( 1 + (0.5 - 0.866i)T \) |
| 29 | \( 1 + (-0.5 + 0.866i)T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 + (0.5 + 0.866i)T \) |
| 41 | \( 1 + (-0.5 - 0.866i)T \) |
| 43 | \( 1 + (0.5 - 0.866i)T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 + (0.5 - 0.866i)T \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 + T \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 + (0.5 - 0.866i)T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 + (0.5 - 0.866i)T \) |
| 89 | \( 1 + (-0.5 - 0.866i)T \) |
| 97 | \( 1 + (0.5 - 0.866i)T \) |
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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
−25.42651141626657500239798670027, −24.782641698682895410725119992685, −23.532838623147381235164524655968, −22.86442779173130581207012818204, −21.31051821421897942482612456666, −20.80725477095978614301776335855, −19.7775722881571833693907245956, −18.99588101375694693411737268198, −18.01699587392931080882213117893, −17.361685291752814900785060177705, −16.40454493521745701126861629386, −15.35875902337042038013203272938, −14.79647798096610300895054318461, −13.15592029691364929678535970461, −12.3316049515598151367922958925, −11.18788637502465448177593477424, −10.267187057789013548555621856593, −9.58472264639141652965827006537, −8.21836806469695458046528723929, −7.72659664903244246441934431635, −6.42353740328047845559127146181, −5.465240868548750617601626380, −3.785623670593405827404168169328, −2.49378609777426484246325922101, −1.26269482897681912002713583224,
0.76348293408459853018640275375, 2.27598636880795436964486003322, 3.3653104673916634490882894604, 5.03424675974209429754510994361, 6.32773569739068958959672955195, 7.1487826598589263342768689799, 8.389141049327466856274372135658, 9.006168022886328707305385176198, 10.159985882006666804623227335132, 11.08722866457559694462186656456, 11.805629221400044387895011064065, 13.10147121777073921547730930189, 14.20138689695441192490971479911, 15.351145973389531627470283925234, 16.26820774450642618943666644126, 16.84808135621073050280771343136, 18.03782629244256684633952184605, 18.7711310573133294513998537600, 19.41157211809700430417504929981, 20.67460412262162674441488346700, 21.14426701226040714744321317971, 22.29147558772776199547511773066, 23.68514645958731235377312194356, 24.20859291777744372181710188122, 25.319001305719427082774941010183