| L(s) = 1 | − 1.34e5·3-s − 1.48e7·5-s − 6.55e8·7-s + 7.59e9·9-s − 3.00e10·11-s + 8.04e11·13-s + 2.00e12·15-s + 5.45e12·17-s − 1.16e13·19-s + 8.81e13·21-s − 2.46e14·23-s − 2.55e14·25-s + 3.85e14·27-s − 4.15e14·29-s − 5.30e15·31-s + 4.03e15·33-s + 9.76e15·35-s − 1.98e16·37-s − 1.08e17·39-s − 1.02e17·41-s + 1.62e17·43-s − 1.13e17·45-s + 2.43e17·47-s − 1.28e17·49-s − 7.33e17·51-s + 6.13e17·53-s + 4.47e17·55-s + ⋯ |
| L(s) = 1 | − 1.31·3-s − 0.681·5-s − 0.877·7-s + 0.725·9-s − 0.349·11-s + 1.61·13-s + 0.895·15-s + 0.656·17-s − 0.437·19-s + 1.15·21-s − 1.24·23-s − 0.535·25-s + 0.360·27-s − 0.183·29-s − 1.16·31-s + 0.458·33-s + 0.598·35-s − 0.679·37-s − 2.12·39-s − 1.19·41-s + 1.14·43-s − 0.494·45-s + 0.674·47-s − 0.229·49-s − 0.862·51-s + 0.481·53-s + 0.238·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(\approx\) |
\(0.2356823682\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2356823682\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + 1.34e5T + 1.04e10T^{2} \) |
| 5 | \( 1 + 1.48e7T + 4.76e14T^{2} \) |
| 7 | \( 1 + 6.55e8T + 5.58e17T^{2} \) |
| 11 | \( 1 + 3.00e10T + 7.40e21T^{2} \) |
| 13 | \( 1 - 8.04e11T + 2.47e23T^{2} \) |
| 17 | \( 1 - 5.45e12T + 6.90e25T^{2} \) |
| 19 | \( 1 + 1.16e13T + 7.14e26T^{2} \) |
| 23 | \( 1 + 2.46e14T + 3.94e28T^{2} \) |
| 29 | \( 1 + 4.15e14T + 5.13e30T^{2} \) |
| 31 | \( 1 + 5.30e15T + 2.08e31T^{2} \) |
| 37 | \( 1 + 1.98e16T + 8.55e32T^{2} \) |
| 41 | \( 1 + 1.02e17T + 7.38e33T^{2} \) |
| 43 | \( 1 - 1.62e17T + 2.00e34T^{2} \) |
| 47 | \( 1 - 2.43e17T + 1.30e35T^{2} \) |
| 53 | \( 1 - 6.13e17T + 1.62e36T^{2} \) |
| 59 | \( 1 + 5.51e18T + 1.54e37T^{2} \) |
| 61 | \( 1 + 4.86e18T + 3.10e37T^{2} \) |
| 67 | \( 1 + 2.24e19T + 2.22e38T^{2} \) |
| 71 | \( 1 + 4.96e19T + 7.52e38T^{2} \) |
| 73 | \( 1 - 4.50e19T + 1.34e39T^{2} \) |
| 79 | \( 1 + 1.08e20T + 7.08e39T^{2} \) |
| 83 | \( 1 + 8.71e19T + 1.99e40T^{2} \) |
| 89 | \( 1 + 3.00e20T + 8.65e40T^{2} \) |
| 97 | \( 1 + 7.52e20T + 5.27e41T^{2} \) |
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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
−10.99286418717304232751917506518, −10.18368149224030344047837114288, −8.724359350466661724834790565750, −7.46790430819288499302442451377, −6.20620835980361348698791493890, −5.68389781246694158999308120184, −4.19766250624948705084312659456, −3.29168869249509422800008596348, −1.49558803060013583988744143171, −0.23024277884568397940738798130,
0.23024277884568397940738798130, 1.49558803060013583988744143171, 3.29168869249509422800008596348, 4.19766250624948705084312659456, 5.68389781246694158999308120184, 6.20620835980361348698791493890, 7.46790430819288499302442451377, 8.724359350466661724834790565750, 10.18368149224030344047837114288, 10.99286418717304232751917506518
