| L(s) = 1 | + 0.472·2-s − 1.64·3-s − 1.77·4-s − 3.20·5-s − 0.777·6-s + 4.06·7-s − 1.78·8-s − 0.290·9-s − 1.51·10-s − 11-s + 2.92·12-s − 6.29·13-s + 1.91·14-s + 5.28·15-s + 2.70·16-s + 2.28·17-s − 0.137·18-s + 4.41·19-s + 5.70·20-s − 6.68·21-s − 0.472·22-s + 4.18·23-s + 2.93·24-s + 5.30·25-s − 2.97·26-s + 5.41·27-s − 7.21·28-s + ⋯ |
| L(s) = 1 | + 0.334·2-s − 0.950·3-s − 0.888·4-s − 1.43·5-s − 0.317·6-s + 1.53·7-s − 0.631·8-s − 0.0967·9-s − 0.479·10-s − 0.301·11-s + 0.844·12-s − 1.74·13-s + 0.513·14-s + 1.36·15-s + 0.677·16-s + 0.554·17-s − 0.0323·18-s + 1.01·19-s + 1.27·20-s − 1.45·21-s − 0.100·22-s + 0.873·23-s + 0.599·24-s + 1.06·25-s − 0.583·26-s + 1.04·27-s − 1.36·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6849531823\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6849531823\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 - 0.472T + 2T^{2} \) |
| 3 | \( 1 + 1.64T + 3T^{2} \) |
| 5 | \( 1 + 3.20T + 5T^{2} \) |
| 7 | \( 1 - 4.06T + 7T^{2} \) |
| 13 | \( 1 + 6.29T + 13T^{2} \) |
| 17 | \( 1 - 2.28T + 17T^{2} \) |
| 19 | \( 1 - 4.41T + 19T^{2} \) |
| 23 | \( 1 - 4.18T + 23T^{2} \) |
| 29 | \( 1 - 2.32T + 29T^{2} \) |
| 31 | \( 1 - 4.52T + 31T^{2} \) |
| 37 | \( 1 + 8.64T + 37T^{2} \) |
| 41 | \( 1 - 4.05T + 41T^{2} \) |
| 43 | \( 1 - 11.3T + 43T^{2} \) |
| 47 | \( 1 + 0.746T + 47T^{2} \) |
| 53 | \( 1 + 9.63T + 53T^{2} \) |
| 59 | \( 1 - 2.52T + 59T^{2} \) |
| 67 | \( 1 - 10.5T + 67T^{2} \) |
| 71 | \( 1 + 14.4T + 71T^{2} \) |
| 73 | \( 1 - 5.36T + 73T^{2} \) |
| 79 | \( 1 - 10.4T + 79T^{2} \) |
| 83 | \( 1 + 1.73T + 83T^{2} \) |
| 89 | \( 1 + 13.7T + 89T^{2} \) |
| 97 | \( 1 - 0.00104T + 97T^{2} \) |
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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
−10.82291065594446266807118866356, −9.726039592200463369272015903557, −8.555093825210907433190053263865, −7.85666427066269104540483020614, −7.19093747142897173871852463907, −5.51758129667386906774805259259, −4.93659778194879004310063596875, −4.41690416927015960861022698808, −3.02205789614756793643663827720, −0.70395390335304220986904615402,
0.70395390335304220986904615402, 3.02205789614756793643663827720, 4.41690416927015960861022698808, 4.93659778194879004310063596875, 5.51758129667386906774805259259, 7.19093747142897173871852463907, 7.85666427066269104540483020614, 8.555093825210907433190053263865, 9.726039592200463369272015903557, 10.82291065594446266807118866356
