| L(s) = 1 | − 1.74·2-s − 1.96·3-s + 1.04·4-s − 3.28·5-s + 3.42·6-s + 7-s + 1.66·8-s + 0.841·9-s + 5.73·10-s + 1.91·11-s − 2.05·12-s − 3.31·13-s − 1.74·14-s + 6.43·15-s − 4.99·16-s − 1.86·17-s − 1.46·18-s + 8.48·19-s − 3.44·20-s − 1.96·21-s − 3.33·22-s + 2.57·23-s − 3.25·24-s + 5.79·25-s + 5.78·26-s + 4.22·27-s + 1.04·28-s + ⋯ |
| L(s) = 1 | − 1.23·2-s − 1.13·3-s + 0.524·4-s − 1.46·5-s + 1.39·6-s + 0.377·7-s + 0.587·8-s + 0.280·9-s + 1.81·10-s + 0.576·11-s − 0.593·12-s − 0.919·13-s − 0.466·14-s + 1.66·15-s − 1.24·16-s − 0.452·17-s − 0.346·18-s + 1.94·19-s − 0.769·20-s − 0.427·21-s − 0.711·22-s + 0.536·23-s − 0.664·24-s + 1.15·25-s + 1.13·26-s + 0.814·27-s + 0.198·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3936958827\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3936958827\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - T \) |
| 859 | \( 1 + T \) |
| good | 2 | \( 1 + 1.74T + 2T^{2} \) |
| 3 | \( 1 + 1.96T + 3T^{2} \) |
| 5 | \( 1 + 3.28T + 5T^{2} \) |
| 11 | \( 1 - 1.91T + 11T^{2} \) |
| 13 | \( 1 + 3.31T + 13T^{2} \) |
| 17 | \( 1 + 1.86T + 17T^{2} \) |
| 19 | \( 1 - 8.48T + 19T^{2} \) |
| 23 | \( 1 - 2.57T + 23T^{2} \) |
| 29 | \( 1 - 0.829T + 29T^{2} \) |
| 31 | \( 1 + 2.16T + 31T^{2} \) |
| 37 | \( 1 - 2.77T + 37T^{2} \) |
| 41 | \( 1 - 10.8T + 41T^{2} \) |
| 43 | \( 1 + 2.93T + 43T^{2} \) |
| 47 | \( 1 - 11.9T + 47T^{2} \) |
| 53 | \( 1 - 2.58T + 53T^{2} \) |
| 59 | \( 1 + 8.63T + 59T^{2} \) |
| 61 | \( 1 - 3.85T + 61T^{2} \) |
| 67 | \( 1 - 2.23T + 67T^{2} \) |
| 71 | \( 1 + 8.59T + 71T^{2} \) |
| 73 | \( 1 - 2.15T + 73T^{2} \) |
| 79 | \( 1 + 6.24T + 79T^{2} \) |
| 83 | \( 1 - 9.36T + 83T^{2} \) |
| 89 | \( 1 - 9.54T + 89T^{2} \) |
| 97 | \( 1 - 16.4T + 97T^{2} \) |
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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
−7.944901635713154552064804801561, −7.39359092733584165350207005410, −7.13096340830733649269584614833, −6.06024133805677453622347953663, −5.08956767274092304009974640537, −4.60640428622456932453993171074, −3.77951246917678316549955546049, −2.62306367489311195304645758498, −1.15114235451302114160229005615, −0.51959505386814995330812700618,
0.51959505386814995330812700618, 1.15114235451302114160229005615, 2.62306367489311195304645758498, 3.77951246917678316549955546049, 4.60640428622456932453993171074, 5.08956767274092304009974640537, 6.06024133805677453622347953663, 7.13096340830733649269584614833, 7.39359092733584165350207005410, 7.944901635713154552064804801561
