L(s) = 1 | + (−0.707 + 0.707i)3-s + 7-s − i·9-s + (0.707 − 0.707i)11-s + (−0.707 + 0.707i)13-s − i·17-s + (0.707 + 0.707i)19-s + (−0.707 + 0.707i)21-s + 23-s + (0.707 + 0.707i)27-s + (0.707 + 0.707i)29-s − 31-s + i·33-s + (−0.707 − 0.707i)37-s − i·39-s + ⋯ |
L(s) = 1 | + (−0.707 + 0.707i)3-s + 7-s − i·9-s + (0.707 − 0.707i)11-s + (−0.707 + 0.707i)13-s − i·17-s + (0.707 + 0.707i)19-s + (−0.707 + 0.707i)21-s + 23-s + (0.707 + 0.707i)27-s + (0.707 + 0.707i)29-s − 31-s + i·33-s + (−0.707 − 0.707i)37-s − i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.681 + 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.681 + 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9101689850 + 0.3960599601i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9101689850 + 0.3960599601i\) |
\(L(1)\) |
\(\approx\) |
\(0.9307089166 + 0.2308418525i\) |
\(L(1)\) |
\(\approx\) |
\(0.9307089166 + 0.2308418525i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (-0.707 + 0.707i)T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + (0.707 - 0.707i)T \) |
| 13 | \( 1 + (-0.707 + 0.707i)T \) |
| 17 | \( 1 - iT \) |
| 19 | \( 1 + (0.707 + 0.707i)T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + (0.707 + 0.707i)T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 + (-0.707 - 0.707i)T \) |
| 41 | \( 1 + iT \) |
| 43 | \( 1 + (0.707 + 0.707i)T \) |
| 47 | \( 1 - iT \) |
| 53 | \( 1 + (-0.707 - 0.707i)T \) |
| 59 | \( 1 + (0.707 - 0.707i)T \) |
| 61 | \( 1 + (-0.707 - 0.707i)T \) |
| 67 | \( 1 + (0.707 - 0.707i)T \) |
| 71 | \( 1 + iT \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 + (0.707 - 0.707i)T \) |
| 89 | \( 1 - iT \) |
| 97 | \( 1 - iT \) |
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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
−27.59533133494779176717527218787, −27.130680368759292004122128828790, −25.40249019661867909653934988382, −24.677101564265189855952110412200, −23.92237206584886463439306979317, −22.75478013566759636648183901874, −22.15778004891226362550678544665, −20.756247126382788245761102148007, −19.79550064766797918332238355053, −18.63798738165399379893945393341, −17.56902225142179235751192314212, −17.247531760160840037635618297186, −15.751854811414704242879619450066, −14.5642152528688991566300065510, −13.54557928310559914778068074063, −12.28978246799140132826021877584, −11.60601705691310793566410543032, −10.54085357515506456937353340585, −9.117904911451569125756618882871, −7.62939125895061455810439256511, −7.00101161715341240687292288546, −5.43179179565065155000769772192, −4.64167992884722649442906956267, −2.5402723979605657848810833565, −1.12370116796876730111395763647,
1.45868257978250108641184873182, 3.530773444132406594607545375852, 4.67485417028504886219423317308, 5.685494188128314379734287038845, 6.94973935488019096893675409397, 8.46664669199954949379154105112, 9.5251859056442567911175100783, 10.78943558935228141898574537798, 11.51356235816057314164650727801, 12.49998516235453340363437410930, 14.25914144294554027685091715853, 14.81401985251042175900980218141, 16.203469765259166250551083422899, 16.9798950088707752585489789392, 17.808884617789974818133173180772, 19.02905933360468029958465077724, 20.2676559661811969639506066564, 21.44590729694975573710082786102, 21.808224933879595074275047237810, 23.07495555592713017557178153200, 24.011139996471189872301310820550, 24.82352256882932393682465446086, 26.37816417484005399250202489286, 27.09916483923375163040190591859, 27.76812729835312519120152392208