| L(s) = 1 | + 904.·2-s − 1.27e6·4-s − 9.76e6·5-s − 5.27e8·7-s − 3.05e9·8-s − 8.83e9·10-s − 2.65e10·11-s − 2.99e11·13-s − 4.77e11·14-s − 7.79e10·16-s − 5.63e12·17-s − 2.99e13·19-s + 1.24e13·20-s − 2.39e13·22-s − 2.68e14·23-s + 9.53e13·25-s − 2.70e14·26-s + 6.75e14·28-s − 2.29e15·29-s + 9.85e12·31-s + 6.33e15·32-s − 5.09e15·34-s + 5.15e15·35-s − 4.70e16·37-s − 2.70e16·38-s + 2.98e16·40-s + 5.27e16·41-s + ⋯ |
| L(s) = 1 | + 0.624·2-s − 0.610·4-s − 0.447·5-s − 0.706·7-s − 1.00·8-s − 0.279·10-s − 0.308·11-s − 0.601·13-s − 0.440·14-s − 0.0177·16-s − 0.677·17-s − 1.11·19-s + 0.272·20-s − 0.192·22-s − 1.35·23-s + 0.199·25-s − 0.375·26-s + 0.430·28-s − 1.01·29-s + 0.00215·31-s + 0.994·32-s − 0.423·34-s + 0.315·35-s − 1.60·37-s − 0.699·38-s + 0.449·40-s + 0.614·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(\approx\) |
\(0.3373801234\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3373801234\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 9.76e6T \) |
| good | 2 | \( 1 - 904.T + 2.09e6T^{2} \) |
| 7 | \( 1 + 5.27e8T + 5.58e17T^{2} \) |
| 11 | \( 1 + 2.65e10T + 7.40e21T^{2} \) |
| 13 | \( 1 + 2.99e11T + 2.47e23T^{2} \) |
| 17 | \( 1 + 5.63e12T + 6.90e25T^{2} \) |
| 19 | \( 1 + 2.99e13T + 7.14e26T^{2} \) |
| 23 | \( 1 + 2.68e14T + 3.94e28T^{2} \) |
| 29 | \( 1 + 2.29e15T + 5.13e30T^{2} \) |
| 31 | \( 1 - 9.85e12T + 2.08e31T^{2} \) |
| 37 | \( 1 + 4.70e16T + 8.55e32T^{2} \) |
| 41 | \( 1 - 5.27e16T + 7.38e33T^{2} \) |
| 43 | \( 1 + 1.73e17T + 2.00e34T^{2} \) |
| 47 | \( 1 - 3.55e17T + 1.30e35T^{2} \) |
| 53 | \( 1 - 1.59e18T + 1.62e36T^{2} \) |
| 59 | \( 1 - 7.01e17T + 1.54e37T^{2} \) |
| 61 | \( 1 - 6.72e16T + 3.10e37T^{2} \) |
| 67 | \( 1 + 1.91e19T + 2.22e38T^{2} \) |
| 71 | \( 1 - 3.18e19T + 7.52e38T^{2} \) |
| 73 | \( 1 - 3.26e19T + 1.34e39T^{2} \) |
| 79 | \( 1 - 4.87e19T + 7.08e39T^{2} \) |
| 83 | \( 1 + 9.61e19T + 1.99e40T^{2} \) |
| 89 | \( 1 - 1.96e20T + 8.65e40T^{2} \) |
| 97 | \( 1 + 1.19e21T + 5.27e41T^{2} \) |
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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
−11.96074283593725619367122696111, −10.44939744300976616184603740482, −9.298404705597883102847862730836, −8.216298746334524550534206733170, −6.76376768866397102012494928622, −5.57474196846547401488620095613, −4.37755596931862454633534939895, −3.52582439359354811509256135224, −2.21983737381598459435588831124, −0.22887907292494402519623239194,
0.22887907292494402519623239194, 2.21983737381598459435588831124, 3.52582439359354811509256135224, 4.37755596931862454633534939895, 5.57474196846547401488620095613, 6.76376768866397102012494928622, 8.216298746334524550534206733170, 9.298404705597883102847862730836, 10.44939744300976616184603740482, 11.96074283593725619367122696111
