| L(s) = 1 | − 2.19·2-s − 1.71·3-s + 2.81·4-s − 2.23·5-s + 3.75·6-s − 3.04·7-s − 1.78·8-s − 0.0678·9-s + 4.90·10-s − 1.23·11-s − 4.81·12-s + 4.62·13-s + 6.68·14-s + 3.82·15-s − 1.71·16-s − 2.82·17-s + 0.148·18-s − 2.42·19-s − 6.28·20-s + 5.21·21-s + 2.71·22-s − 1.68·23-s + 3.04·24-s + 0.00115·25-s − 10.1·26-s + 5.25·27-s − 8.56·28-s + ⋯ |
| L(s) = 1 | − 1.55·2-s − 0.988·3-s + 1.40·4-s − 1.00·5-s + 1.53·6-s − 1.15·7-s − 0.629·8-s − 0.0226·9-s + 1.55·10-s − 0.372·11-s − 1.38·12-s + 1.28·13-s + 1.78·14-s + 0.988·15-s − 0.429·16-s − 0.683·17-s + 0.0350·18-s − 0.556·19-s − 1.40·20-s + 1.13·21-s + 0.578·22-s − 0.351·23-s + 0.622·24-s + 0.000230·25-s − 1.98·26-s + 1.01·27-s − 1.61·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 29 | \( 1 - T \) |
| 139 | \( 1 - T \) |
| good | 2 | \( 1 + 2.19T + 2T^{2} \) |
| 3 | \( 1 + 1.71T + 3T^{2} \) |
| 5 | \( 1 + 2.23T + 5T^{2} \) |
| 7 | \( 1 + 3.04T + 7T^{2} \) |
| 11 | \( 1 + 1.23T + 11T^{2} \) |
| 13 | \( 1 - 4.62T + 13T^{2} \) |
| 17 | \( 1 + 2.82T + 17T^{2} \) |
| 19 | \( 1 + 2.42T + 19T^{2} \) |
| 23 | \( 1 + 1.68T + 23T^{2} \) |
| 31 | \( 1 + 3.83T + 31T^{2} \) |
| 37 | \( 1 + 11.7T + 37T^{2} \) |
| 41 | \( 1 - 5.64T + 41T^{2} \) |
| 43 | \( 1 - 1.28T + 43T^{2} \) |
| 47 | \( 1 - 2.94T + 47T^{2} \) |
| 53 | \( 1 - 6.00T + 53T^{2} \) |
| 59 | \( 1 + 2.00T + 59T^{2} \) |
| 61 | \( 1 + 0.627T + 61T^{2} \) |
| 67 | \( 1 + 0.175T + 67T^{2} \) |
| 71 | \( 1 - 10.5T + 71T^{2} \) |
| 73 | \( 1 - 8.86T + 73T^{2} \) |
| 79 | \( 1 - 12.9T + 79T^{2} \) |
| 83 | \( 1 - 6.51T + 83T^{2} \) |
| 89 | \( 1 - 6.91T + 89T^{2} \) |
| 97 | \( 1 + 6.03T + 97T^{2} \) |
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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
−8.235287433981698456781682760745, −7.44897795930083425979838632425, −6.63711618087035954606360434210, −6.27904927582517266144905975232, −5.29947333544592317217613973552, −4.13846560404673941429526991304, −3.37229661588937345210812039047, −2.13597947676627688289609365725, −0.72413633878497003053265343352, 0,
0.72413633878497003053265343352, 2.13597947676627688289609365725, 3.37229661588937345210812039047, 4.13846560404673941429526991304, 5.29947333544592317217613973552, 6.27904927582517266144905975232, 6.63711618087035954606360434210, 7.44897795930083425979838632425, 8.235287433981698456781682760745
