L(s) = 1 | + (0.5 − 0.866i)3-s − i·5-s + (−0.866 + 0.5i)7-s + (−0.5 − 0.866i)9-s + (0.866 + 0.5i)11-s + (−0.866 − 0.5i)15-s + (0.5 + 0.866i)17-s + (0.866 − 0.5i)19-s + i·21-s + (−0.5 + 0.866i)23-s − 25-s − 27-s + (−0.5 + 0.866i)29-s − i·31-s + (0.866 − 0.5i)33-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)3-s − i·5-s + (−0.866 + 0.5i)7-s + (−0.5 − 0.866i)9-s + (0.866 + 0.5i)11-s + (−0.866 − 0.5i)15-s + (0.5 + 0.866i)17-s + (0.866 − 0.5i)19-s + i·21-s + (−0.5 + 0.866i)23-s − 25-s − 27-s + (−0.5 + 0.866i)29-s − i·31-s + (0.866 − 0.5i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.477 - 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.477 - 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8113134835 - 0.4824666976i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8113134835 - 0.4824666976i\) |
\(L(1)\) |
\(\approx\) |
\(1.008432410 - 0.3743251013i\) |
\(L(1)\) |
\(\approx\) |
\(1.008432410 - 0.3743251013i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 13 | \( 1 \) |
good | 3 | \( 1 + (0.5 - 0.866i)T \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 + (-0.866 + 0.5i)T \) |
| 11 | \( 1 + (0.866 + 0.5i)T \) |
| 17 | \( 1 + (0.5 + 0.866i)T \) |
| 19 | \( 1 + (0.866 - 0.5i)T \) |
| 23 | \( 1 + (-0.5 + 0.866i)T \) |
| 29 | \( 1 + (-0.5 + 0.866i)T \) |
| 31 | \( 1 - iT \) |
| 37 | \( 1 + (-0.866 - 0.5i)T \) |
| 41 | \( 1 + (0.866 + 0.5i)T \) |
| 43 | \( 1 + (-0.5 - 0.866i)T \) |
| 47 | \( 1 - iT \) |
| 53 | \( 1 + T \) |
| 59 | \( 1 + (-0.866 + 0.5i)T \) |
| 61 | \( 1 + (-0.5 - 0.866i)T \) |
| 67 | \( 1 + (-0.866 - 0.5i)T \) |
| 71 | \( 1 + (0.866 - 0.5i)T \) |
| 73 | \( 1 + iT \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 - iT \) |
| 89 | \( 1 + (-0.866 - 0.5i)T \) |
| 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
−33.4612076135661110716322163537, −32.598112542238019905809802899992, −31.546601904742011694915609326631, −30.272912657030889623951250997479, −29.27129032025964943269196890358, −27.6693708030242225991026658229, −26.65459561868199966084668066735, −25.9801985375403338083760459633, −24.74207554536101707691268189942, −22.68485433299358497072920095247, −22.36714500205458038703503728994, −20.855447554307605177437142518361, −19.669457907846057272051578894564, −18.665482802413952056756565210259, −16.84455902307985961113052664605, −15.85100221675302379412875489627, −14.48911727298974742556966853600, −13.69846051657300168303630676056, −11.60865316285847333600719433102, −10.27988488578117742584337442162, −9.387847552481165577054408399814, −7.59196732415173634910989196993, −6.06661818175461017362427215795, −3.96713462095470882962957348290, −2.911774947049750184837037758224,
1.594344909186465607486496135145, 3.543778027803707331971337932293, 5.65865003222743779439604456608, 7.118871054388509400601730555779, 8.65479939217735953144787239716, 9.56385130074579473761258416431, 11.99075933269504376238149852323, 12.68016620188670958722716708569, 13.89275246709670516198497570793, 15.39301724092766888996275662592, 16.798358020688816216639308250243, 18.03781345044416577382605048980, 19.50832942047579863056275947718, 20.03465887703240906528573038656, 21.601805436819209155834469841313, 23.092798173623892469495352099024, 24.27684419558324045336901634080, 25.16634257370934945713832025785, 26.03883110551415798928535961649, 27.84852352326918570443574975224, 28.77087060674976115076006935860, 29.88244729957315520371571420395, 31.09865760863963456472685008357, 32.06074664895349836424746949139, 32.93283785916200077966705338440