L(s) = 1 | + (0.909 − 0.415i)3-s + (−0.540 + 0.841i)7-s + (0.654 − 0.755i)9-s + (−0.142 − 0.989i)11-s + (−0.540 − 0.841i)13-s + (−0.281 − 0.959i)17-s + (0.959 + 0.281i)19-s + (−0.142 + 0.989i)21-s + (0.281 − 0.959i)27-s + (−0.959 + 0.281i)29-s + (0.415 − 0.909i)31-s + (−0.540 − 0.841i)33-s + (0.755 + 0.654i)37-s + (−0.841 − 0.540i)39-s + (−0.654 − 0.755i)41-s + ⋯ |
L(s) = 1 | + (0.909 − 0.415i)3-s + (−0.540 + 0.841i)7-s + (0.654 − 0.755i)9-s + (−0.142 − 0.989i)11-s + (−0.540 − 0.841i)13-s + (−0.281 − 0.959i)17-s + (0.959 + 0.281i)19-s + (−0.142 + 0.989i)21-s + (0.281 − 0.959i)27-s + (−0.959 + 0.281i)29-s + (0.415 − 0.909i)31-s + (−0.540 − 0.841i)33-s + (0.755 + 0.654i)37-s + (−0.841 − 0.540i)39-s + (−0.654 − 0.755i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0426 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0426 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.206638121 - 1.156271876i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.206638121 - 1.156271876i\) |
\(L(1)\) |
\(\approx\) |
\(1.229223804 - 0.3778012931i\) |
\(L(1)\) |
\(\approx\) |
\(1.229223804 - 0.3778012931i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 \) |
good | 3 | \( 1 + (0.909 - 0.415i)T \) |
| 7 | \( 1 + (-0.540 + 0.841i)T \) |
| 11 | \( 1 + (-0.142 - 0.989i)T \) |
| 13 | \( 1 + (-0.540 - 0.841i)T \) |
| 17 | \( 1 + (-0.281 - 0.959i)T \) |
| 19 | \( 1 + (0.959 + 0.281i)T \) |
| 29 | \( 1 + (-0.959 + 0.281i)T \) |
| 31 | \( 1 + (0.415 - 0.909i)T \) |
| 37 | \( 1 + (0.755 + 0.654i)T \) |
| 41 | \( 1 + (-0.654 - 0.755i)T \) |
| 43 | \( 1 + (-0.909 + 0.415i)T \) |
| 47 | \( 1 - iT \) |
| 53 | \( 1 + (-0.540 + 0.841i)T \) |
| 59 | \( 1 + (0.841 - 0.540i)T \) |
| 61 | \( 1 + (0.415 - 0.909i)T \) |
| 67 | \( 1 + (0.989 + 0.142i)T \) |
| 71 | \( 1 + (-0.142 + 0.989i)T \) |
| 73 | \( 1 + (0.281 - 0.959i)T \) |
| 79 | \( 1 + (0.841 - 0.540i)T \) |
| 83 | \( 1 + (-0.755 - 0.654i)T \) |
| 89 | \( 1 + (0.415 + 0.909i)T \) |
| 97 | \( 1 + (0.755 - 0.654i)T \) |
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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
−22.01468698412864833087672816111, −21.19221087895045572555194912196, −20.366905605768177643275154442057, −19.77861296720134295330385926384, −19.24021235384370902415974359824, −18.18620513646441726013401451237, −17.199420729001002392119440259694, −16.446763984533793737593548185526, −15.66188665180905251757523880116, −14.82301988243151335494829109810, −14.17676657857498769332501139686, −13.29189447689282693272181788474, −12.69874572414770386544384710464, −11.51645203700119420265550973708, −10.44327081187680161385785034342, −9.785915953028480689105471661726, −9.21692870157984697866677016438, −8.07347572641918904632500300240, −7.28286863519979584611510385759, −6.61722016989447735984735997456, −5.09141839561667326606708316210, −4.265293668220812734799017401356, −3.54947111997939213784238725024, −2.455217635408365001134746712107, −1.48704496013647961608890881802,
0.64253734229090847865093737572, 2.1077112773449771684916862644, 2.97182043832088395486728992627, 3.5246482925461586903402531613, 5.050757704241318801118297302412, 5.89797729170271023602204046950, 6.89810774295684888564391394073, 7.82008227621043677859292035428, 8.52136896096539824095694616876, 9.43027195736807236753484598279, 9.96936718001082550756199244058, 11.366901632165400161189251057343, 12.09982710712554896985860690991, 13.04650269984083616673145765087, 13.560353082096308515677654629848, 14.494439285136440532575797369268, 15.322084774969261615738752740231, 15.90691545945640964646174428971, 16.87700992028840827529961879888, 18.125622239484781765718056894106, 18.58697437900849022805659162104, 19.23142560798145115483734836686, 20.21357023265997802242810908167, 20.61511127490974062529628642064, 21.84387534941589681435788468834