L(s) = 1 | − 0.879·2-s − 1.22·4-s − 1.34·5-s + 2.83·8-s + 1.18·10-s + 1.65·11-s + 3.36·13-s − 0.0418·16-s − 0.467·17-s + 3.22·19-s + 1.65·20-s − 1.45·22-s + 8.94·23-s − 3.18·25-s − 2.96·26-s + 6.26·29-s − 9.23·31-s − 5.63·32-s + 0.411·34-s + 9.23·37-s − 2.83·38-s − 3.82·40-s − 3.41·41-s − 4.41·43-s − 2.02·44-s − 7.86·46-s − 9.35·47-s + ⋯ |
L(s) = 1 | − 0.621·2-s − 0.613·4-s − 0.602·5-s + 1.00·8-s + 0.374·10-s + 0.498·11-s + 0.934·13-s − 0.0104·16-s − 0.113·17-s + 0.740·19-s + 0.369·20-s − 0.309·22-s + 1.86·23-s − 0.636·25-s − 0.581·26-s + 1.16·29-s − 1.65·31-s − 0.996·32-s + 0.0705·34-s + 1.51·37-s − 0.460·38-s − 0.604·40-s − 0.532·41-s − 0.672·43-s − 0.305·44-s − 1.15·46-s − 1.36·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.062479769\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.062479769\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + 0.879T + 2T^{2} \) |
| 5 | \( 1 + 1.34T + 5T^{2} \) |
| 11 | \( 1 - 1.65T + 11T^{2} \) |
| 13 | \( 1 - 3.36T + 13T^{2} \) |
| 17 | \( 1 + 0.467T + 17T^{2} \) |
| 19 | \( 1 - 3.22T + 19T^{2} \) |
| 23 | \( 1 - 8.94T + 23T^{2} \) |
| 29 | \( 1 - 6.26T + 29T^{2} \) |
| 31 | \( 1 + 9.23T + 31T^{2} \) |
| 37 | \( 1 - 9.23T + 37T^{2} \) |
| 41 | \( 1 + 3.41T + 41T^{2} \) |
| 43 | \( 1 + 4.41T + 43T^{2} \) |
| 47 | \( 1 + 9.35T + 47T^{2} \) |
| 53 | \( 1 + 0.573T + 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 + 7.63T + 61T^{2} \) |
| 67 | \( 1 - 0.596T + 67T^{2} \) |
| 71 | \( 1 + 0.554T + 71T^{2} \) |
| 73 | \( 1 + 2.04T + 73T^{2} \) |
| 79 | \( 1 + 2.40T + 79T^{2} \) |
| 83 | \( 1 - 15.0T + 83T^{2} \) |
| 89 | \( 1 + 9.08T + 89T^{2} \) |
| 97 | \( 1 - 1.89T + 97T^{2} \) |
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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
−8.541079537757555426369837875937, −7.86134618667314776414849375605, −7.19494929518110329553662179102, −6.40115179086270890211068944386, −5.36279926417163303112258394961, −4.65605091171053168362806434722, −3.81767676503564471828293940856, −3.14663059564583488423574395184, −1.58602013408515275323041461300, −0.70689527052284079729243613953,
0.70689527052284079729243613953, 1.58602013408515275323041461300, 3.14663059564583488423574395184, 3.81767676503564471828293940856, 4.65605091171053168362806434722, 5.36279926417163303112258394961, 6.40115179086270890211068944386, 7.19494929518110329553662179102, 7.86134618667314776414849375605, 8.541079537757555426369837875937