L(s) = 1 | + (0.563 − 1.29i)2-s + (−1.36 − 1.46i)4-s − 3.07·5-s − 3.99i·7-s + (−2.66 + 0.949i)8-s + (−1.73 + 3.99i)10-s + 1.89i·11-s − 1.06i·13-s + (−5.17 − 2.24i)14-s + (−0.267 + 3.99i)16-s − 7.08i·17-s + 3.73·19-s + (4.20 + 4.49i)20-s + (2.46 + 1.06i)22-s + 0.824·23-s + ⋯ |
L(s) = 1 | + (0.398 − 0.917i)2-s + (−0.683 − 0.730i)4-s − 1.37·5-s − 1.50i·7-s + (−0.941 + 0.335i)8-s + (−0.547 + 1.26i)10-s + 0.572i·11-s − 0.296i·13-s + (−1.38 − 0.600i)14-s + (−0.0669 + 0.997i)16-s − 1.71i·17-s + 0.856·19-s + (0.939 + 1.00i)20-s + (0.525 + 0.227i)22-s + 0.171·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.941 + 0.335i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.941 + 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.157132 - 0.908785i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.157132 - 0.908785i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.563 + 1.29i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 3.07T + 5T^{2} \) |
| 7 | \( 1 + 3.99iT - 7T^{2} \) |
| 11 | \( 1 - 1.89iT - 11T^{2} \) |
| 13 | \( 1 + 1.06iT - 13T^{2} \) |
| 17 | \( 1 + 7.08iT - 17T^{2} \) |
| 19 | \( 1 - 3.73T + 19T^{2} \) |
| 23 | \( 1 - 0.824T + 23T^{2} \) |
| 29 | \( 1 - 4.50T + 29T^{2} \) |
| 31 | \( 1 - 5.84iT - 31T^{2} \) |
| 37 | \( 1 + 6.91iT - 37T^{2} \) |
| 41 | \( 1 + 3.79iT - 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 - 6.97T + 47T^{2} \) |
| 53 | \( 1 + 10.6T + 53T^{2} \) |
| 59 | \( 1 - 1.89iT - 59T^{2} \) |
| 61 | \( 1 - 1.06iT - 61T^{2} \) |
| 67 | \( 1 + 9.19T + 67T^{2} \) |
| 71 | \( 1 - 10.6T + 71T^{2} \) |
| 73 | \( 1 - 5.92T + 73T^{2} \) |
| 79 | \( 1 - 6.12iT - 79T^{2} \) |
| 83 | \( 1 + 10.3iT - 83T^{2} \) |
| 89 | \( 1 + 13.6iT - 89T^{2} \) |
| 97 | \( 1 + 11.3T + 97T^{2} \) |
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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
−11.86013762084490049507837754005, −11.04496623957535397595158019408, −10.25447685104577004244687905488, −9.176266309946643644059967635200, −7.72848249981788528792689224149, −7.02537682218175160446851979800, −5.01243412768810945954081389713, −4.14782519357128457744940176652, −3.12991891071461417768027143380, −0.72174568451695627049820039506,
3.12623223825844541732146445101, 4.29348342391439892155982542404, 5.58448750037658199332841466311, 6.52525116457653613053433315151, 7.978138294220392001728567308736, 8.364092606005528757519908255651, 9.427844527481960599995296683109, 11.18445054792200275386750641724, 12.05276313728417987228687123190, 12.57836259405508190500195904109