| L(s) = 1 | + 8.90e6·5-s + 5.87e8·7-s + 1.42e11·11-s + 1.49e11·13-s − 9.95e12·17-s − 5.02e12·19-s − 3.25e14·23-s − 3.97e14·25-s − 1.07e15·29-s + 3.98e15·31-s + 5.23e15·35-s − 3.35e16·37-s − 4.54e16·41-s − 8.38e16·43-s + 5.34e16·47-s − 2.13e17·49-s − 1.14e18·53-s + 1.26e18·55-s + 5.97e18·59-s + 4.12e17·61-s + 1.32e18·65-s − 2.42e19·67-s − 7.25e16·71-s − 2.97e19·73-s + 8.36e19·77-s − 5.78e19·79-s + 2.48e20·83-s + ⋯ |
| L(s) = 1 | + 0.407·5-s + 0.786·7-s + 1.65·11-s + 0.299·13-s − 1.19·17-s − 0.188·19-s − 1.63·23-s − 0.833·25-s − 0.472·29-s + 0.873·31-s + 0.320·35-s − 1.14·37-s − 0.529·41-s − 0.591·43-s + 0.148·47-s − 0.381·49-s − 0.896·53-s + 0.674·55-s + 1.52·59-s + 0.0740·61-s + 0.122·65-s − 1.62·67-s − 0.00264·71-s − 0.810·73-s + 1.30·77-s − 0.687·79-s + 1.75·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 8.90e6T + 4.76e14T^{2} \) |
| 7 | \( 1 - 5.87e8T + 5.58e17T^{2} \) |
| 11 | \( 1 - 1.42e11T + 7.40e21T^{2} \) |
| 13 | \( 1 - 1.49e11T + 2.47e23T^{2} \) |
| 17 | \( 1 + 9.95e12T + 6.90e25T^{2} \) |
| 19 | \( 1 + 5.02e12T + 7.14e26T^{2} \) |
| 23 | \( 1 + 3.25e14T + 3.94e28T^{2} \) |
| 29 | \( 1 + 1.07e15T + 5.13e30T^{2} \) |
| 31 | \( 1 - 3.98e15T + 2.08e31T^{2} \) |
| 37 | \( 1 + 3.35e16T + 8.55e32T^{2} \) |
| 41 | \( 1 + 4.54e16T + 7.38e33T^{2} \) |
| 43 | \( 1 + 8.38e16T + 2.00e34T^{2} \) |
| 47 | \( 1 - 5.34e16T + 1.30e35T^{2} \) |
| 53 | \( 1 + 1.14e18T + 1.62e36T^{2} \) |
| 59 | \( 1 - 5.97e18T + 1.54e37T^{2} \) |
| 61 | \( 1 - 4.12e17T + 3.10e37T^{2} \) |
| 67 | \( 1 + 2.42e19T + 2.22e38T^{2} \) |
| 71 | \( 1 + 7.25e16T + 7.52e38T^{2} \) |
| 73 | \( 1 + 2.97e19T + 1.34e39T^{2} \) |
| 79 | \( 1 + 5.78e19T + 7.08e39T^{2} \) |
| 83 | \( 1 - 2.48e20T + 1.99e40T^{2} \) |
| 89 | \( 1 - 5.56e19T + 8.65e40T^{2} \) |
| 97 | \( 1 + 8.39e20T + 5.27e41T^{2} \) |
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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
−10.13541484037796682477682326976, −9.062102360248290359317296426281, −8.173991681947699330774394010367, −6.77593525709775374052024815140, −5.94281136601750307735235540879, −4.56498557739984571892659630078, −3.71659645046990881627389106338, −2.05150015848198013902444283728, −1.44588857549832529339111158188, 0,
1.44588857549832529339111158188, 2.05150015848198013902444283728, 3.71659645046990881627389106338, 4.56498557739984571892659630078, 5.94281136601750307735235540879, 6.77593525709775374052024815140, 8.173991681947699330774394010367, 9.062102360248290359317296426281, 10.13541484037796682477682326976
