L(s) = 1 | + (−0.0581 − 0.998i)2-s + (−0.993 + 0.116i)4-s + (0.973 + 0.230i)5-s + (0.597 − 0.802i)7-s + (0.173 + 0.984i)8-s + (0.173 − 0.984i)10-s + (−0.686 − 0.727i)11-s + (0.893 − 0.448i)13-s + (−0.835 − 0.549i)14-s + (0.973 − 0.230i)16-s + (−0.939 + 0.342i)17-s + (−0.939 − 0.342i)19-s + (−0.993 − 0.116i)20-s + (−0.686 + 0.727i)22-s + (0.597 + 0.802i)23-s + ⋯ |
L(s) = 1 | + (−0.0581 − 0.998i)2-s + (−0.993 + 0.116i)4-s + (0.973 + 0.230i)5-s + (0.597 − 0.802i)7-s + (0.173 + 0.984i)8-s + (0.173 − 0.984i)10-s + (−0.686 − 0.727i)11-s + (0.893 − 0.448i)13-s + (−0.835 − 0.549i)14-s + (0.973 − 0.230i)16-s + (−0.939 + 0.342i)17-s + (−0.939 − 0.342i)19-s + (−0.993 − 0.116i)20-s + (−0.686 + 0.727i)22-s + (0.597 + 0.802i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0968 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0968 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7404552842 - 0.6719273433i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7404552842 - 0.6719273433i\) |
\(L(1)\) |
\(\approx\) |
\(0.9109849642 - 0.5334985951i\) |
\(L(1)\) |
\(\approx\) |
\(0.9109849642 - 0.5334985951i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
good | 2 | \( 1 + (-0.0581 - 0.998i)T \) |
| 5 | \( 1 + (0.973 + 0.230i)T \) |
| 7 | \( 1 + (0.597 - 0.802i)T \) |
| 11 | \( 1 + (-0.686 - 0.727i)T \) |
| 13 | \( 1 + (0.893 - 0.448i)T \) |
| 17 | \( 1 + (-0.939 + 0.342i)T \) |
| 19 | \( 1 + (-0.939 - 0.342i)T \) |
| 23 | \( 1 + (0.597 + 0.802i)T \) |
| 29 | \( 1 + (-0.835 + 0.549i)T \) |
| 31 | \( 1 + (0.396 - 0.918i)T \) |
| 37 | \( 1 + (0.766 + 0.642i)T \) |
| 41 | \( 1 + (-0.0581 + 0.998i)T \) |
| 43 | \( 1 + (-0.286 + 0.957i)T \) |
| 47 | \( 1 + (0.396 + 0.918i)T \) |
| 53 | \( 1 + (-0.5 + 0.866i)T \) |
| 59 | \( 1 + (-0.686 + 0.727i)T \) |
| 61 | \( 1 + (-0.993 - 0.116i)T \) |
| 67 | \( 1 + (-0.835 - 0.549i)T \) |
| 71 | \( 1 + (0.173 - 0.984i)T \) |
| 73 | \( 1 + (0.173 + 0.984i)T \) |
| 79 | \( 1 + (-0.0581 - 0.998i)T \) |
| 83 | \( 1 + (-0.0581 - 0.998i)T \) |
| 89 | \( 1 + (0.173 + 0.984i)T \) |
| 97 | \( 1 + (0.973 - 0.230i)T \) |
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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
−31.40584372602693181979932031726, −30.51929028243888528111954612517, −28.70977781342254806902253798402, −28.15176078437410164478693117810, −26.75957638663801881788245752485, −25.647617675307911703010514858294, −24.99081748843174119827314310643, −23.982397026812508466688358373335, −22.80750285417904185149793608329, −21.60155832017538735677225405010, −20.72490049914285390275871871251, −18.70240842034910860705532497120, −17.99688902292953567794904107235, −17.030283772829479434354187269, −15.73843611353678655587488796359, −14.750039837009186405351653188166, −13.581362917040254131951456922802, −12.58118065513609780647048507000, −10.646258577308296930929432481651, −9.212298094002598992335445944088, −8.41001201841857183147924176460, −6.75949259448805605852981760594, −5.61853700754692715941359835036, −4.541165499558567760715165613372, −2.05450028644898398116301178421,
1.4479441812529719786381235513, 2.9755007393240368736737084078, 4.56299343697567634235595656035, 6.0208524936554592081327124008, 8.002902726759443833341305732842, 9.254851094640615781039281659083, 10.70314371361620846084698749085, 11.07127927059037263451464529943, 13.217623195794901364045866712743, 13.459654905027872434755464453932, 14.91861148245749097408555779542, 16.89227762950962207962065836504, 17.79231805840609575078113863472, 18.69715229093568804550706553446, 20.09214673353839363884491756576, 21.05209738322340936715023610149, 21.752986510152455334600781344107, 23.06569252977329530745503425278, 24.09616315024363280596614451370, 25.77291462574095172992217237699, 26.58668505864200477749867140844, 27.72508206793947145378391167277, 28.83739070478102770425556552827, 29.74709619129973350041102187197, 30.39828911869058810625361852054