L(s) = 1 | + 2.68·2-s + 2.18·3-s + 5.20·4-s + 2.69·5-s + 5.86·6-s − 4.16·7-s + 8.61·8-s + 1.77·9-s + 7.24·10-s − 5.77·11-s + 11.3·12-s + 13-s − 11.1·14-s + 5.89·15-s + 12.7·16-s − 5.44·17-s + 4.77·18-s + 3.24·19-s + 14.0·20-s − 9.10·21-s − 15.5·22-s − 6.13·23-s + 18.8·24-s + 2.27·25-s + 2.68·26-s − 2.67·27-s − 21.7·28-s + ⋯ |
L(s) = 1 | + 1.89·2-s + 1.26·3-s + 2.60·4-s + 1.20·5-s + 2.39·6-s − 1.57·7-s + 3.04·8-s + 0.592·9-s + 2.29·10-s − 1.74·11-s + 3.28·12-s + 0.277·13-s − 2.98·14-s + 1.52·15-s + 3.18·16-s − 1.31·17-s + 1.12·18-s + 0.744·19-s + 3.14·20-s − 1.98·21-s − 3.30·22-s − 1.27·23-s + 3.84·24-s + 0.455·25-s + 0.526·26-s − 0.514·27-s − 4.10·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(7.580970846\) |
\(L(\frac12)\) |
\(\approx\) |
\(7.580970846\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 - T \) |
| 103 | \( 1 + T \) |
good | 2 | \( 1 - 2.68T + 2T^{2} \) |
| 3 | \( 1 - 2.18T + 3T^{2} \) |
| 5 | \( 1 - 2.69T + 5T^{2} \) |
| 7 | \( 1 + 4.16T + 7T^{2} \) |
| 11 | \( 1 + 5.77T + 11T^{2} \) |
| 17 | \( 1 + 5.44T + 17T^{2} \) |
| 19 | \( 1 - 3.24T + 19T^{2} \) |
| 23 | \( 1 + 6.13T + 23T^{2} \) |
| 29 | \( 1 - 8.69T + 29T^{2} \) |
| 31 | \( 1 + 4.43T + 31T^{2} \) |
| 37 | \( 1 - 0.754T + 37T^{2} \) |
| 41 | \( 1 - 6.21T + 41T^{2} \) |
| 43 | \( 1 - 7.50T + 43T^{2} \) |
| 47 | \( 1 - 3.05T + 47T^{2} \) |
| 53 | \( 1 + 2.04T + 53T^{2} \) |
| 59 | \( 1 - 14.8T + 59T^{2} \) |
| 61 | \( 1 + 2.20T + 61T^{2} \) |
| 67 | \( 1 - 6.75T + 67T^{2} \) |
| 71 | \( 1 - 6.00T + 71T^{2} \) |
| 73 | \( 1 + 7.24T + 73T^{2} \) |
| 79 | \( 1 + 11.7T + 79T^{2} \) |
| 83 | \( 1 + 13.6T + 83T^{2} \) |
| 89 | \( 1 - 7.43T + 89T^{2} \) |
| 97 | \( 1 + 2.44T + 97T^{2} \) |
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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
−9.868933980957623313150601769067, −8.827596624816463831898401994051, −7.72256496114876715600385129844, −6.87432730483331151644965962060, −6.00055609969599977380626871542, −5.56309923231489425236446827484, −4.33941708820410644012512894150, −3.37392774680155980630366280856, −2.55055084242296355231947440507, −2.29845929145320381313851825461,
2.29845929145320381313851825461, 2.55055084242296355231947440507, 3.37392774680155980630366280856, 4.33941708820410644012512894150, 5.56309923231489425236446827484, 6.00055609969599977380626871542, 6.87432730483331151644965962060, 7.72256496114876715600385129844, 8.827596624816463831898401994051, 9.868933980957623313150601769067