| L(s) = 1 | − 2.24·2-s − 1.44·3-s + 3.04·4-s + 5-s + 3.24·6-s − 7-s − 2.35·8-s − 0.911·9-s − 2.24·10-s + 0.780·11-s − 4.40·12-s − 1.64·13-s + 2.24·14-s − 1.44·15-s − 0.801·16-s + 17-s + 2.04·18-s + 7.09·19-s + 3.04·20-s + 1.44·21-s − 1.75·22-s − 8.70·23-s + 3.40·24-s + 25-s + 3.69·26-s + 5.65·27-s − 3.04·28-s + ⋯ |
| L(s) = 1 | − 1.58·2-s − 0.834·3-s + 1.52·4-s + 0.447·5-s + 1.32·6-s − 0.377·7-s − 0.833·8-s − 0.303·9-s − 0.710·10-s + 0.235·11-s − 1.27·12-s − 0.455·13-s + 0.600·14-s − 0.373·15-s − 0.200·16-s + 0.242·17-s + 0.482·18-s + 1.62·19-s + 0.681·20-s + 0.315·21-s − 0.373·22-s − 1.81·23-s + 0.695·24-s + 0.200·25-s + 0.724·26-s + 1.08·27-s − 0.576·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 595 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 595 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 + 2.24T + 2T^{2} \) |
| 3 | \( 1 + 1.44T + 3T^{2} \) |
| 11 | \( 1 - 0.780T + 11T^{2} \) |
| 13 | \( 1 + 1.64T + 13T^{2} \) |
| 19 | \( 1 - 7.09T + 19T^{2} \) |
| 23 | \( 1 + 8.70T + 23T^{2} \) |
| 29 | \( 1 - 3.34T + 29T^{2} \) |
| 31 | \( 1 - 4.02T + 31T^{2} \) |
| 37 | \( 1 + 6.33T + 37T^{2} \) |
| 41 | \( 1 + 12.4T + 41T^{2} \) |
| 43 | \( 1 - 6.94T + 43T^{2} \) |
| 47 | \( 1 + 11.9T + 47T^{2} \) |
| 53 | \( 1 + 8.51T + 53T^{2} \) |
| 59 | \( 1 + 4.66T + 59T^{2} \) |
| 61 | \( 1 + 14.3T + 61T^{2} \) |
| 67 | \( 1 - 5.33T + 67T^{2} \) |
| 71 | \( 1 + 8.81T + 71T^{2} \) |
| 73 | \( 1 + 3.97T + 73T^{2} \) |
| 79 | \( 1 - 3.66T + 79T^{2} \) |
| 83 | \( 1 + 3.91T + 83T^{2} \) |
| 89 | \( 1 + 5.71T + 89T^{2} \) |
| 97 | \( 1 - 7.88T + 97T^{2} \) |
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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
−10.00929896708534700366298427827, −9.623144699669398990087648664046, −8.566839864505368292085832071518, −7.74010063753880648259115649257, −6.72676497307860562235364043916, −5.98652859883397980132051747883, −4.89404734112732022071348771132, −3.02998050545538397213492350204, −1.51687457881933789286040076288, 0,
1.51687457881933789286040076288, 3.02998050545538397213492350204, 4.89404734112732022071348771132, 5.98652859883397980132051747883, 6.72676497307860562235364043916, 7.74010063753880648259115649257, 8.566839864505368292085832071518, 9.623144699669398990087648664046, 10.00929896708534700366298427827
