L(s) = 1 | + (−0.900 + 0.433i)2-s + (0.623 − 0.781i)4-s + (−0.733 − 0.680i)5-s + (−0.222 + 0.974i)8-s + (0.955 + 0.294i)10-s + (0.826 + 0.563i)11-s + (0.826 + 0.563i)13-s + (−0.222 − 0.974i)16-s + (−0.988 + 0.149i)17-s + (−0.5 − 0.866i)19-s + (−0.988 + 0.149i)20-s + (−0.988 − 0.149i)22-s + (0.365 + 0.930i)23-s + (0.0747 + 0.997i)25-s + (−0.988 − 0.149i)26-s + ⋯ |
L(s) = 1 | + (−0.900 + 0.433i)2-s + (0.623 − 0.781i)4-s + (−0.733 − 0.680i)5-s + (−0.222 + 0.974i)8-s + (0.955 + 0.294i)10-s + (0.826 + 0.563i)11-s + (0.826 + 0.563i)13-s + (−0.222 − 0.974i)16-s + (−0.988 + 0.149i)17-s + (−0.5 − 0.866i)19-s + (−0.988 + 0.149i)20-s + (−0.988 − 0.149i)22-s + (0.365 + 0.930i)23-s + (0.0747 + 0.997i)25-s + (−0.988 − 0.149i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.904 + 0.427i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.904 + 0.427i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7461937261 + 0.1675197818i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7461937261 + 0.1675197818i\) |
\(L(1)\) |
\(\approx\) |
\(0.6826866988 + 0.08396463852i\) |
\(L(1)\) |
\(\approx\) |
\(0.6826866988 + 0.08396463852i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.900 + 0.433i)T \) |
| 5 | \( 1 + (-0.733 - 0.680i)T \) |
| 11 | \( 1 + (0.826 + 0.563i)T \) |
| 13 | \( 1 + (0.826 + 0.563i)T \) |
| 17 | \( 1 + (-0.988 + 0.149i)T \) |
| 19 | \( 1 + (-0.5 - 0.866i)T \) |
| 23 | \( 1 + (0.365 + 0.930i)T \) |
| 29 | \( 1 + (-0.988 + 0.149i)T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 + (0.365 - 0.930i)T \) |
| 41 | \( 1 + (0.955 - 0.294i)T \) |
| 43 | \( 1 + (0.955 + 0.294i)T \) |
| 47 | \( 1 + (-0.900 + 0.433i)T \) |
| 53 | \( 1 + (0.365 + 0.930i)T \) |
| 59 | \( 1 + (-0.222 - 0.974i)T \) |
| 61 | \( 1 + (0.623 + 0.781i)T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 + (0.623 - 0.781i)T \) |
| 73 | \( 1 + (0.826 - 0.563i)T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 + (0.826 - 0.563i)T \) |
| 89 | \( 1 + (0.0747 + 0.997i)T \) |
| 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
−24.23403683776666348264390292522, −22.8463598660474257545742362996, −22.37019462440605052565277048106, −21.240618798219498498156681810597, −20.37644406793602078402003526973, −19.568758663851471933545722210, −18.82794702984745124708504300414, −18.20462176296968633038139390864, −17.16898007445304216244923224965, −16.29360560504775067672746869583, −15.47535700577123107860230106328, −14.570939682141952484090774819894, −13.28831814126106626092997862183, −12.26674884375606398524316662511, −11.2702885985506173316037900962, −10.87414389868667176024163503931, −9.78625966391850102814899751315, −8.59165033457922906006185428199, −8.080624514975977023759575247091, −6.83724052414532267272627035270, −6.187131326298146053906869994867, −4.18405600374474704847600543655, −3.41458352601863135411222634675, −2.31224757347518183487197666256, −0.80693918545159013279144303091,
0.961195760347226971852115396949, 2.11573906758137473432325225783, 3.87788325778029631384198518778, 4.83909767988714778243901080418, 6.17091861054590914809254752395, 7.04242723821933461775962126568, 7.9687404789160436954468413664, 9.07741352091457967436457550403, 9.3045962791031241876971261031, 10.98405044741180068138315775414, 11.397764367888356462759485925011, 12.56004268701278305367140547767, 13.686517236289514886879373013150, 14.910773086513163480697297511920, 15.58124514435742529432290871226, 16.33340049383901817097873721558, 17.251525396171545696146312675424, 17.86617827711820951470140117221, 19.16923196570187259226132815854, 19.57955254806342975578611357510, 20.42307212325562717455953452311, 21.300456723010709857099873268162, 22.69993159368239583397700189987, 23.51140762509313272880427403243, 24.25955916687519693560421323718