| L(s) = 1 | − 1.13·2-s − 0.720·4-s + 2.38·5-s + 4.08·7-s + 3.07·8-s − 2.70·10-s + 3.85·11-s + 13-s − 4.62·14-s − 2.03·16-s + 2.15·17-s − 0.284·19-s − 1.72·20-s − 4.35·22-s + 3.58·23-s + 0.700·25-s − 1.13·26-s − 2.94·28-s + 2.97·29-s − 10.0·31-s − 3.84·32-s − 2.44·34-s + 9.76·35-s − 7.86·37-s + 0.322·38-s + 7.34·40-s + 5.57·41-s + ⋯ |
| L(s) = 1 | − 0.799·2-s − 0.360·4-s + 1.06·5-s + 1.54·7-s + 1.08·8-s − 0.853·10-s + 1.16·11-s + 0.277·13-s − 1.23·14-s − 0.509·16-s + 0.523·17-s − 0.0653·19-s − 0.384·20-s − 0.929·22-s + 0.748·23-s + 0.140·25-s − 0.221·26-s − 0.556·28-s + 0.552·29-s − 1.79·31-s − 0.680·32-s − 0.418·34-s + 1.64·35-s − 1.29·37-s + 0.0522·38-s + 1.16·40-s + 0.870·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1053 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1053 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.504685772\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.504685772\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 + 1.13T + 2T^{2} \) |
| 5 | \( 1 - 2.38T + 5T^{2} \) |
| 7 | \( 1 - 4.08T + 7T^{2} \) |
| 11 | \( 1 - 3.85T + 11T^{2} \) |
| 17 | \( 1 - 2.15T + 17T^{2} \) |
| 19 | \( 1 + 0.284T + 19T^{2} \) |
| 23 | \( 1 - 3.58T + 23T^{2} \) |
| 29 | \( 1 - 2.97T + 29T^{2} \) |
| 31 | \( 1 + 10.0T + 31T^{2} \) |
| 37 | \( 1 + 7.86T + 37T^{2} \) |
| 41 | \( 1 - 5.57T + 41T^{2} \) |
| 43 | \( 1 + 2.82T + 43T^{2} \) |
| 47 | \( 1 + 8.42T + 47T^{2} \) |
| 53 | \( 1 + 4.34T + 53T^{2} \) |
| 59 | \( 1 - 9.58T + 59T^{2} \) |
| 61 | \( 1 + 7.68T + 61T^{2} \) |
| 67 | \( 1 - 1.46T + 67T^{2} \) |
| 71 | \( 1 + 8.81T + 71T^{2} \) |
| 73 | \( 1 - 15.7T + 73T^{2} \) |
| 79 | \( 1 - 9.82T + 79T^{2} \) |
| 83 | \( 1 - 0.672T + 83T^{2} \) |
| 89 | \( 1 - 3.26T + 89T^{2} \) |
| 97 | \( 1 + 8.87T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.688921674486921714632543819059, −9.114174723141862155088670121439, −8.451421475011083970882065068073, −7.64466740183839270937585995625, −6.67959966036787184145883723711, −5.49414736518430186892959611656, −4.84826286875653249520811291477, −3.74315898260817906515459422723, −1.88382195789424486516607426066, −1.26910033145183415341381326288,
1.26910033145183415341381326288, 1.88382195789424486516607426066, 3.74315898260817906515459422723, 4.84826286875653249520811291477, 5.49414736518430186892959611656, 6.67959966036787184145883723711, 7.64466740183839270937585995625, 8.451421475011083970882065068073, 9.114174723141862155088670121439, 9.688921674486921714632543819059
