| L(s) = 1 | − 2.61·2-s + 1.81·3-s + 4.84·4-s + 3.74·5-s − 4.75·6-s + 0.406·7-s − 7.43·8-s + 0.304·9-s − 9.80·10-s + 11-s + 8.80·12-s − 3.01·13-s − 1.06·14-s + 6.81·15-s + 9.76·16-s − 6.49·17-s − 0.796·18-s − 1.64·19-s + 18.1·20-s + 0.738·21-s − 2.61·22-s − 1.93·23-s − 13.5·24-s + 9.04·25-s + 7.88·26-s − 4.90·27-s + 1.96·28-s + ⋯ |
| L(s) = 1 | − 1.84·2-s + 1.04·3-s + 2.42·4-s + 1.67·5-s − 1.94·6-s + 0.153·7-s − 2.62·8-s + 0.101·9-s − 3.09·10-s + 0.301·11-s + 2.54·12-s − 0.835·13-s − 0.284·14-s + 1.75·15-s + 2.44·16-s − 1.57·17-s − 0.187·18-s − 0.377·19-s + 4.05·20-s + 0.161·21-s − 0.557·22-s − 0.404·23-s − 2.75·24-s + 1.80·25-s + 1.54·26-s − 0.943·27-s + 0.371·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 - T \) |
| 547 | \( 1 - T \) |
| good | 2 | \( 1 + 2.61T + 2T^{2} \) |
| 3 | \( 1 - 1.81T + 3T^{2} \) |
| 5 | \( 1 - 3.74T + 5T^{2} \) |
| 7 | \( 1 - 0.406T + 7T^{2} \) |
| 13 | \( 1 + 3.01T + 13T^{2} \) |
| 17 | \( 1 + 6.49T + 17T^{2} \) |
| 19 | \( 1 + 1.64T + 19T^{2} \) |
| 23 | \( 1 + 1.93T + 23T^{2} \) |
| 29 | \( 1 + 7.98T + 29T^{2} \) |
| 31 | \( 1 + 2.09T + 31T^{2} \) |
| 37 | \( 1 + 4.51T + 37T^{2} \) |
| 41 | \( 1 - 8.72T + 41T^{2} \) |
| 43 | \( 1 + 11.3T + 43T^{2} \) |
| 47 | \( 1 + 1.52T + 47T^{2} \) |
| 53 | \( 1 + 5.40T + 53T^{2} \) |
| 59 | \( 1 - 5.97T + 59T^{2} \) |
| 61 | \( 1 - 13.7T + 61T^{2} \) |
| 67 | \( 1 - 7.92T + 67T^{2} \) |
| 71 | \( 1 - 12.1T + 71T^{2} \) |
| 73 | \( 1 + 6.94T + 73T^{2} \) |
| 79 | \( 1 - 7.28T + 79T^{2} \) |
| 83 | \( 1 - 8.02T + 83T^{2} \) |
| 89 | \( 1 - 0.679T + 89T^{2} \) |
| 97 | \( 1 + 14.7T + 97T^{2} \) |
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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
−8.126380968778194264196630845132, −7.11347292213751743671705512329, −6.68450912713972960012147859771, −5.92445436005406152603643017000, −5.05340535273736906660314039042, −3.64502653965387600636356347904, −2.44328479189190905102015523941, −2.21013020336724721534198203479, −1.57364847846550094244004760861, 0,
1.57364847846550094244004760861, 2.21013020336724721534198203479, 2.44328479189190905102015523941, 3.64502653965387600636356347904, 5.05340535273736906660314039042, 5.92445436005406152603643017000, 6.68450912713972960012147859771, 7.11347292213751743671705512329, 8.126380968778194264196630845132
