L(s) = 1 | + (−0.707 − 0.707i)2-s + (0.130 − 0.991i)3-s + i·4-s + (−0.991 − 0.130i)5-s + (−0.793 + 0.608i)6-s + (−0.991 + 0.130i)7-s + (0.707 − 0.707i)8-s + (−0.965 − 0.258i)9-s + (0.608 + 0.793i)10-s + (−0.923 − 0.382i)11-s + (0.991 + 0.130i)12-s + (−0.866 + 0.5i)13-s + (0.793 + 0.608i)14-s + (−0.258 + 0.965i)15-s − 16-s + ⋯ |
L(s) = 1 | + (−0.707 − 0.707i)2-s + (0.130 − 0.991i)3-s + i·4-s + (−0.991 − 0.130i)5-s + (−0.793 + 0.608i)6-s + (−0.991 + 0.130i)7-s + (0.707 − 0.707i)8-s + (−0.965 − 0.258i)9-s + (0.608 + 0.793i)10-s + (−0.923 − 0.382i)11-s + (0.991 + 0.130i)12-s + (−0.866 + 0.5i)13-s + (0.793 + 0.608i)14-s + (−0.258 + 0.965i)15-s − 16-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.811 - 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.811 - 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3206743551 - 0.1033587206i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3206743551 - 0.1033587206i\) |
\(L(1)\) |
\(\approx\) |
\(0.3953208654 - 0.2423396845i\) |
\(L(1)\) |
\(\approx\) |
\(0.3953208654 - 0.2423396845i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 \) |
| 43 | \( 1 \) |
good | 2 | \( 1 + (-0.707 - 0.707i)T \) |
| 3 | \( 1 + (0.130 - 0.991i)T \) |
| 5 | \( 1 + (-0.991 - 0.130i)T \) |
| 7 | \( 1 + (-0.991 + 0.130i)T \) |
| 11 | \( 1 + (-0.923 - 0.382i)T \) |
| 13 | \( 1 + (-0.866 + 0.5i)T \) |
| 19 | \( 1 + (0.965 - 0.258i)T \) |
| 23 | \( 1 + (-0.793 + 0.608i)T \) |
| 29 | \( 1 + (-0.608 + 0.793i)T \) |
| 31 | \( 1 + (0.130 - 0.991i)T \) |
| 37 | \( 1 + (0.130 - 0.991i)T \) |
| 41 | \( 1 + (-0.382 + 0.923i)T \) |
| 47 | \( 1 - iT \) |
| 53 | \( 1 + (-0.965 + 0.258i)T \) |
| 59 | \( 1 + (-0.707 + 0.707i)T \) |
| 61 | \( 1 + (0.991 - 0.130i)T \) |
| 67 | \( 1 + (0.5 - 0.866i)T \) |
| 71 | \( 1 + (0.793 + 0.608i)T \) |
| 73 | \( 1 + (0.608 - 0.793i)T \) |
| 79 | \( 1 + (0.130 + 0.991i)T \) |
| 83 | \( 1 + (0.965 - 0.258i)T \) |
| 89 | \( 1 + (0.866 + 0.5i)T \) |
| 97 | \( 1 + (0.382 + 0.923i)T \) |
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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
−22.60847481462224264377657897850, −22.188736522718785988936380028, −20.534035338267466805893568927170, −20.18558968977595345366636452799, −19.33290386747485414579547246626, −18.636308226127085084415986081216, −17.54958939950120467724455053016, −16.68132648117875795834259242552, −15.79706789499674292557002902842, −15.702586910826241146544829996480, −14.75913283912640941854183983448, −13.92251335283198002527032201462, −12.63409494678162288145719704982, −11.595156738848771301485487472389, −10.48113890673063546491798318885, −10.049786702009155236809452366811, −9.25541656205209160051951102734, −8.12170463697585865855818907242, −7.60729777051714450626125540416, −6.532765085300587147762301553286, −5.3892072020734625991528474964, −4.63702842677587760083708276363, −3.474976014147613717639213430104, −2.512509440030712287316038797087, −0.33129796962108790824266564904,
0.65752462940379460708769216313, 2.114589936882506179053554619955, 3.00884566417207902450525885893, 3.75059621773191668142319906759, 5.24533782083225955217181178712, 6.62217506171683544419050205530, 7.50871820577025114709670606368, 7.9283087401444922117425633215, 9.02765786354422338777248321572, 9.73961247879150484947111943349, 10.98105577144956957825872037097, 11.73745264355642399530068010984, 12.39339968463152949592162098582, 13.05953813168980965697217856130, 13.8683421138881796027371384048, 15.24466574023554304623275582570, 16.20675933272782545812723226988, 16.7485151353541027915018662723, 17.9213301350843429365625895857, 18.6397965055929252862022976959, 19.14480996943100155585621729864, 19.93294899073935407092803103811, 20.278310734952991584785444596112, 21.622096153357542755037640693921, 22.40938812696431393354556728106