L(s) = 1 | + (−0.866 − 0.5i)2-s + (−0.866 − 0.5i)3-s + (0.5 + 0.866i)4-s + (0.5 + 0.866i)6-s + (0.866 − 0.5i)7-s − i·8-s + (0.5 + 0.866i)9-s + (0.5 − 0.866i)11-s − i·12-s − 14-s + (−0.5 + 0.866i)16-s + (−0.866 + 0.5i)17-s − i·18-s + (−0.5 − 0.866i)19-s − 21-s + (−0.866 + 0.5i)22-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.5i)2-s + (−0.866 − 0.5i)3-s + (0.5 + 0.866i)4-s + (0.5 + 0.866i)6-s + (0.866 − 0.5i)7-s − i·8-s + (0.5 + 0.866i)9-s + (0.5 − 0.866i)11-s − i·12-s − 14-s + (−0.5 + 0.866i)16-s + (−0.866 + 0.5i)17-s − i·18-s + (−0.5 − 0.866i)19-s − 21-s + (−0.866 + 0.5i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.886 - 0.462i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.886 - 0.462i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1430851451 - 0.5836864112i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1430851451 - 0.5836864112i\) |
\(L(1)\) |
\(\approx\) |
\(0.4814200058 - 0.3061342289i\) |
\(L(1)\) |
\(\approx\) |
\(0.4814200058 - 0.3061342289i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 3 | \( 1 + (-0.866 - 0.5i)T \) |
| 7 | \( 1 + (0.866 - 0.5i)T \) |
| 11 | \( 1 + (0.5 - 0.866i)T \) |
| 17 | \( 1 + (-0.866 + 0.5i)T \) |
| 19 | \( 1 + (-0.5 - 0.866i)T \) |
| 23 | \( 1 + (-0.866 - 0.5i)T \) |
| 29 | \( 1 + (0.5 - 0.866i)T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 + (-0.866 - 0.5i)T \) |
| 41 | \( 1 + (0.5 - 0.866i)T \) |
| 43 | \( 1 + (0.866 - 0.5i)T \) |
| 47 | \( 1 + iT \) |
| 53 | \( 1 + iT \) |
| 59 | \( 1 + (-0.5 - 0.866i)T \) |
| 61 | \( 1 + (-0.5 - 0.866i)T \) |
| 67 | \( 1 + (-0.866 - 0.5i)T \) |
| 71 | \( 1 + (0.5 + 0.866i)T \) |
| 73 | \( 1 - iT \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 - iT \) |
| 89 | \( 1 + (-0.5 + 0.866i)T \) |
| 97 | \( 1 + (0.866 - 0.5i)T \) |
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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
−32.95801610026909597357397195564, −31.42325728496939476378004387011, −29.80577300150334799278415768308, −28.72684696704912443684855371314, −27.68867440864127462561786127855, −27.27829916520530105767620995525, −25.84795024213292963770926586364, −24.65911888015117805103892793369, −23.658923955998136094944011930377, −22.47145671493658974960493067823, −21.11034514857137394562464263144, −19.943843749781430593457944598452, −18.24217291580373679131479492801, −17.69683292848140481816945608231, −16.5409135698675563845899324193, −15.41923308847014282593545380344, −14.49121274773444141092445510142, −12.145460890885197634796437256807, −11.13233649766080621910090023032, −9.94020435762973798068973452161, −8.745600686325298851839618232620, −7.161353555319259545335128787268, −5.83622760604458733970295178423, −4.58768321619286234742794114559, −1.67790011843073376429174685107,
0.47108991964819114539807616418, 1.96613788265165931069765654422, 4.25953145020906238816915988881, 6.26116501409082213448960202704, 7.5474282210882288787743504004, 8.78278991285225465897601098463, 10.64511521528514824812300062829, 11.2300565907423357244214557129, 12.44619183247702835962178820543, 13.80783144085297706752342409664, 15.84730056890954533772186244586, 17.11603932064746316911321942323, 17.67261537478834057788328739483, 18.894752628877394191633363645013, 19.92258485340093942036359284817, 21.35255106537014494638746629077, 22.27162202953397182280235637076, 23.937048182437109049322744006890, 24.62713083235856242023410106007, 26.23192281074283340456557020997, 27.31646233803405221857625899577, 28.1259131526935666944922425547, 29.23196594022520716411706224425, 30.11163406018842243975305450113, 30.83346395362043574301728160049