| L(s) = 1 | + 0.200·2-s − 1.95·4-s − 0.775·7-s − 0.794·8-s − 4.45·11-s − 3.67·13-s − 0.155·14-s + 3.76·16-s + 1.05·17-s + 3.76·19-s − 0.893·22-s + 6.21·23-s − 0.738·26-s + 1.51·28-s − 29-s − 3.96·31-s + 2.34·32-s + 0.211·34-s − 2.64·37-s + 0.754·38-s − 6.67·41-s − 9.96·43-s + 8.72·44-s + 1.24·46-s − 12.6·47-s − 6.39·49-s + 7.21·52-s + ⋯ |
| L(s) = 1 | + 0.141·2-s − 0.979·4-s − 0.292·7-s − 0.280·8-s − 1.34·11-s − 1.02·13-s − 0.0415·14-s + 0.940·16-s + 0.256·17-s + 0.863·19-s − 0.190·22-s + 1.29·23-s − 0.144·26-s + 0.287·28-s − 0.185·29-s − 0.711·31-s + 0.414·32-s + 0.0363·34-s − 0.434·37-s + 0.122·38-s − 1.04·41-s − 1.51·43-s + 1.31·44-s + 0.183·46-s − 1.84·47-s − 0.914·49-s + 0.999·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7961288818\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7961288818\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - 0.200T + 2T^{2} \) |
| 7 | \( 1 + 0.775T + 7T^{2} \) |
| 11 | \( 1 + 4.45T + 11T^{2} \) |
| 13 | \( 1 + 3.67T + 13T^{2} \) |
| 17 | \( 1 - 1.05T + 17T^{2} \) |
| 19 | \( 1 - 3.76T + 19T^{2} \) |
| 23 | \( 1 - 6.21T + 23T^{2} \) |
| 31 | \( 1 + 3.96T + 31T^{2} \) |
| 37 | \( 1 + 2.64T + 37T^{2} \) |
| 41 | \( 1 + 6.67T + 41T^{2} \) |
| 43 | \( 1 + 9.96T + 43T^{2} \) |
| 47 | \( 1 + 12.6T + 47T^{2} \) |
| 53 | \( 1 + 2.87T + 53T^{2} \) |
| 59 | \( 1 - 7.92T + 59T^{2} \) |
| 61 | \( 1 + 2.22T + 61T^{2} \) |
| 67 | \( 1 - 4.12T + 67T^{2} \) |
| 71 | \( 1 + 13.6T + 71T^{2} \) |
| 73 | \( 1 - 8.60T + 73T^{2} \) |
| 79 | \( 1 - 6.97T + 79T^{2} \) |
| 83 | \( 1 - 2.90T + 83T^{2} \) |
| 89 | \( 1 - 4.30T + 89T^{2} \) |
| 97 | \( 1 + 3.82T + 97T^{2} \) |
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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
−8.056519734745020664080131470398, −7.37167576315954373378703927133, −6.68703198374115160085607976101, −5.56565702576643013528397017259, −5.06887066294013350887717148382, −4.71301940956515197924240048806, −3.34213468967141299398206542467, −3.12150596212515588374759967328, −1.82276137764036478220528016386, −0.44169320812110696027517238890,
0.44169320812110696027517238890, 1.82276137764036478220528016386, 3.12150596212515588374759967328, 3.34213468967141299398206542467, 4.71301940956515197924240048806, 5.06887066294013350887717148382, 5.56565702576643013528397017259, 6.68703198374115160085607976101, 7.37167576315954373378703927133, 8.056519734745020664080131470398
