| L(s) = 1 | + 0.843·2-s − 3-s − 1.28·4-s − 2.14·5-s − 0.843·6-s + 4.16·7-s − 2.77·8-s + 9-s − 1.81·10-s + 1.87·11-s + 1.28·12-s − 2.91·13-s + 3.51·14-s + 2.14·15-s + 0.239·16-s + 17-s + 0.843·18-s − 2.76·19-s + 2.76·20-s − 4.16·21-s + 1.57·22-s − 7.37·23-s + 2.77·24-s − 0.390·25-s − 2.45·26-s − 27-s − 5.36·28-s + ⋯ |
| L(s) = 1 | + 0.596·2-s − 0.577·3-s − 0.644·4-s − 0.960·5-s − 0.344·6-s + 1.57·7-s − 0.980·8-s + 0.333·9-s − 0.572·10-s + 0.564·11-s + 0.372·12-s − 0.808·13-s + 0.938·14-s + 0.554·15-s + 0.0599·16-s + 0.242·17-s + 0.198·18-s − 0.634·19-s + 0.618·20-s − 0.908·21-s + 0.336·22-s − 1.53·23-s + 0.566·24-s − 0.0780·25-s − 0.482·26-s − 0.192·27-s − 1.01·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 79 | \( 1 - T \) |
| good | 2 | \( 1 - 0.843T + 2T^{2} \) |
| 5 | \( 1 + 2.14T + 5T^{2} \) |
| 7 | \( 1 - 4.16T + 7T^{2} \) |
| 11 | \( 1 - 1.87T + 11T^{2} \) |
| 13 | \( 1 + 2.91T + 13T^{2} \) |
| 19 | \( 1 + 2.76T + 19T^{2} \) |
| 23 | \( 1 + 7.37T + 23T^{2} \) |
| 29 | \( 1 - 6.85T + 29T^{2} \) |
| 31 | \( 1 - 5.01T + 31T^{2} \) |
| 37 | \( 1 - 3.03T + 37T^{2} \) |
| 41 | \( 1 - 11.8T + 41T^{2} \) |
| 43 | \( 1 + 8.69T + 43T^{2} \) |
| 47 | \( 1 - 7.35T + 47T^{2} \) |
| 53 | \( 1 + 7.09T + 53T^{2} \) |
| 59 | \( 1 + 12.1T + 59T^{2} \) |
| 61 | \( 1 - 2.95T + 61T^{2} \) |
| 67 | \( 1 + 11.9T + 67T^{2} \) |
| 71 | \( 1 + 1.56T + 71T^{2} \) |
| 73 | \( 1 + 4.76T + 73T^{2} \) |
| 83 | \( 1 - 9.50T + 83T^{2} \) |
| 89 | \( 1 + 10.9T + 89T^{2} \) |
| 97 | \( 1 + 10.7T + 97T^{2} \) |
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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
−8.012965171556896980147649543428, −7.54264995970306891580424144702, −6.38054231502568382806848613555, −5.73910442254496593624827650185, −4.68865102803475282302000593898, −4.51605290983628753975886571394, −3.85682927122101287330347896414, −2.57831723799824253709443153872, −1.27106839016531099370962325606, 0,
1.27106839016531099370962325606, 2.57831723799824253709443153872, 3.85682927122101287330347896414, 4.51605290983628753975886571394, 4.68865102803475282302000593898, 5.73910442254496593624827650185, 6.38054231502568382806848613555, 7.54264995970306891580424144702, 8.012965171556896980147649543428
