| L(s) = 1 | + 166.·2-s + 5.90e4·3-s − 2.06e6·4-s + 9.84e6·6-s + 1.02e9·7-s − 6.94e8·8-s + 3.48e9·9-s + 8.41e10·11-s − 1.22e11·12-s + 6.78e11·13-s + 1.70e11·14-s + 4.22e12·16-s − 5.16e12·17-s + 5.81e11·18-s + 5.03e12·19-s + 6.05e13·21-s + 1.40e13·22-s + 3.20e14·23-s − 4.10e13·24-s + 1.13e14·26-s + 2.05e14·27-s − 2.12e15·28-s + 3.64e15·29-s − 4.68e15·31-s + 2.16e15·32-s + 4.96e15·33-s − 8.61e14·34-s + ⋯ |
| L(s) = 1 | + 0.115·2-s + 0.577·3-s − 0.986·4-s + 0.0664·6-s + 1.37·7-s − 0.228·8-s + 0.333·9-s + 0.978·11-s − 0.569·12-s + 1.36·13-s + 0.157·14-s + 0.960·16-s − 0.621·17-s + 0.0383·18-s + 0.188·19-s + 0.791·21-s + 0.112·22-s + 1.61·23-s − 0.132·24-s + 0.157·26-s + 0.192·27-s − 1.35·28-s + 1.61·29-s − 1.02·31-s + 0.339·32-s + 0.564·33-s − 0.0715·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(\approx\) |
\(3.941632437\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.941632437\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 5.90e4T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 166.T + 2.09e6T^{2} \) |
| 7 | \( 1 - 1.02e9T + 5.58e17T^{2} \) |
| 11 | \( 1 - 8.41e10T + 7.40e21T^{2} \) |
| 13 | \( 1 - 6.78e11T + 2.47e23T^{2} \) |
| 17 | \( 1 + 5.16e12T + 6.90e25T^{2} \) |
| 19 | \( 1 - 5.03e12T + 7.14e26T^{2} \) |
| 23 | \( 1 - 3.20e14T + 3.94e28T^{2} \) |
| 29 | \( 1 - 3.64e15T + 5.13e30T^{2} \) |
| 31 | \( 1 + 4.68e15T + 2.08e31T^{2} \) |
| 37 | \( 1 - 5.20e16T + 8.55e32T^{2} \) |
| 41 | \( 1 + 7.94e16T + 7.38e33T^{2} \) |
| 43 | \( 1 - 1.06e17T + 2.00e34T^{2} \) |
| 47 | \( 1 + 1.15e17T + 1.30e35T^{2} \) |
| 53 | \( 1 + 2.45e17T + 1.62e36T^{2} \) |
| 59 | \( 1 + 7.28e18T + 1.54e37T^{2} \) |
| 61 | \( 1 - 1.31e17T + 3.10e37T^{2} \) |
| 67 | \( 1 + 1.83e19T + 2.22e38T^{2} \) |
| 71 | \( 1 + 4.72e19T + 7.52e38T^{2} \) |
| 73 | \( 1 - 3.44e19T + 1.34e39T^{2} \) |
| 79 | \( 1 + 2.81e19T + 7.08e39T^{2} \) |
| 83 | \( 1 - 2.06e20T + 1.99e40T^{2} \) |
| 89 | \( 1 + 8.54e19T + 8.65e40T^{2} \) |
| 97 | \( 1 - 7.33e20T + 5.27e41T^{2} \) |
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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
−10.68411304171208796570103295627, −9.144907125121708919190373542375, −8.711070891330487922725220366164, −7.72438672794775966729631495799, −6.26726451514618390013068074555, −4.88295233783388110949405186036, −4.20297551078695555596735275163, −3.11512587084707751257517672367, −1.52918863604091639414603345868, −0.910362193925667393842712115690,
0.910362193925667393842712115690, 1.52918863604091639414603345868, 3.11512587084707751257517672367, 4.20297551078695555596735275163, 4.88295233783388110949405186036, 6.26726451514618390013068074555, 7.72438672794775966729631495799, 8.711070891330487922725220366164, 9.144907125121708919190373542375, 10.68411304171208796570103295627
