L(s) = 1 | − 1.07i·2-s + i·3-s + 0.842·4-s − 3.36i·5-s + 1.07·6-s − 3.05i·8-s − 9-s − 3.61·10-s + (−3.27 + 0.491i)11-s + 0.842i·12-s + 4.53·13-s + 3.36·15-s − 1.60·16-s + 1.76·17-s + 1.07i·18-s + 7.04·19-s + ⋯ |
L(s) = 1 | − 0.760i·2-s + 0.577i·3-s + 0.421·4-s − 1.50i·5-s + 0.439·6-s − 1.08i·8-s − 0.333·9-s − 1.14·10-s + (−0.988 + 0.148i)11-s + 0.243i·12-s + 1.25·13-s + 0.868·15-s − 0.401·16-s + 0.427·17-s + 0.253i·18-s + 1.61·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.535 + 0.844i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.535 + 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.965202127\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.965202127\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - iT \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (3.27 - 0.491i)T \) |
good | 2 | \( 1 + 1.07iT - 2T^{2} \) |
| 5 | \( 1 + 3.36iT - 5T^{2} \) |
| 13 | \( 1 - 4.53T + 13T^{2} \) |
| 17 | \( 1 - 1.76T + 17T^{2} \) |
| 19 | \( 1 - 7.04T + 19T^{2} \) |
| 23 | \( 1 - 3.35T + 23T^{2} \) |
| 29 | \( 1 + 3.85iT - 29T^{2} \) |
| 31 | \( 1 - 3.22iT - 31T^{2} \) |
| 37 | \( 1 + 9.04T + 37T^{2} \) |
| 41 | \( 1 + 6.67T + 41T^{2} \) |
| 43 | \( 1 + 10.7iT - 43T^{2} \) |
| 47 | \( 1 + 5.21iT - 47T^{2} \) |
| 53 | \( 1 + 11.0T + 53T^{2} \) |
| 59 | \( 1 + 4.52iT - 59T^{2} \) |
| 61 | \( 1 - 3.44T + 61T^{2} \) |
| 67 | \( 1 - 0.302T + 67T^{2} \) |
| 71 | \( 1 - 2.04T + 71T^{2} \) |
| 73 | \( 1 - 10.1T + 73T^{2} \) |
| 79 | \( 1 - 0.790iT - 79T^{2} \) |
| 83 | \( 1 + 1.19T + 83T^{2} \) |
| 89 | \( 1 - 8.59iT - 89T^{2} \) |
| 97 | \( 1 - 5.01iT - 97T^{2} \) |
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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
−9.253099849977403409817826578580, −8.525027580450674893511197140470, −7.76098156750325108405709221102, −6.68276081742950169582453221059, −5.39933437028256356111251842906, −5.11775483426561728502998337946, −3.82319519151935323682419352406, −3.18278742214010895278607778549, −1.78620741204985925465236726756, −0.77498912254526934294813415692,
1.57595807043327386000416423365, 2.94771486306800844972958873742, 3.24475616219444158710814868971, 5.15472353911343830134068850590, 5.87057982905240610243289043175, 6.59210810589697388177013943527, 7.20465606110785004932070789486, 7.79172003762601088533672318151, 8.469880691018743088984007081312, 9.692448774674302138226921997050