| L(s) = 1 | + (−4.85 − 4.85i)2-s + (−79.5 − 191. i)3-s − 464. i·4-s + (1.73e3 − 719. i)5-s + (−546. + 1.31e3i)6-s + (−3.02e3 − 1.25e3i)7-s + (−4.74e3 + 4.74e3i)8-s + (−1.66e4 + 1.66e4i)9-s + (−1.19e4 − 4.93e3i)10-s + (2.32e4 − 5.60e4i)11-s + (−8.92e4 + 3.69e4i)12-s + 7.41e4i·13-s + (8.59e3 + 2.07e4i)14-s + (−2.76e5 − 2.76e5i)15-s − 1.91e5·16-s + (−5.03e3 + 3.44e5i)17-s + ⋯ |
| L(s) = 1 | + (−0.214 − 0.214i)2-s + (−0.566 − 1.36i)3-s − 0.907i·4-s + (1.24 − 0.514i)5-s + (−0.172 + 0.415i)6-s + (−0.475 − 0.197i)7-s + (−0.409 + 0.409i)8-s + (−0.844 + 0.844i)9-s + (−0.377 − 0.156i)10-s + (0.477 − 1.15i)11-s + (−1.24 + 0.514i)12-s + 0.720i·13-s + (0.0598 + 0.144i)14-s + (−1.40 − 1.40i)15-s − 0.732·16-s + (−0.0146 + 0.999i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0904i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.0580497 + 1.28102i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0580497 + 1.28102i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (5.03e3 - 3.44e5i)T \) |
| good | 2 | \( 1 + (4.85 + 4.85i)T + 512iT^{2} \) |
| 3 | \( 1 + (79.5 + 191. i)T + (-1.39e4 + 1.39e4i)T^{2} \) |
| 5 | \( 1 + (-1.73e3 + 719. i)T + (1.38e6 - 1.38e6i)T^{2} \) |
| 7 | \( 1 + (3.02e3 + 1.25e3i)T + (2.85e7 + 2.85e7i)T^{2} \) |
| 11 | \( 1 + (-2.32e4 + 5.60e4i)T + (-1.66e9 - 1.66e9i)T^{2} \) |
| 13 | \( 1 - 7.41e4iT - 1.06e10T^{2} \) |
| 19 | \( 1 + (-4.53e5 - 4.53e5i)T + 3.22e11iT^{2} \) |
| 23 | \( 1 + (-6.08e5 + 1.46e6i)T + (-1.27e12 - 1.27e12i)T^{2} \) |
| 29 | \( 1 + (-3.66e6 + 1.51e6i)T + (1.02e13 - 1.02e13i)T^{2} \) |
| 31 | \( 1 + (3.34e5 + 8.08e5i)T + (-1.86e13 + 1.86e13i)T^{2} \) |
| 37 | \( 1 + (3.64e6 + 8.80e6i)T + (-9.18e13 + 9.18e13i)T^{2} \) |
| 41 | \( 1 + (3.17e7 + 1.31e7i)T + (2.31e14 + 2.31e14i)T^{2} \) |
| 43 | \( 1 + (-1.66e7 + 1.66e7i)T - 5.02e14iT^{2} \) |
| 47 | \( 1 - 3.35e7iT - 1.11e15T^{2} \) |
| 53 | \( 1 + (-4.63e7 - 4.63e7i)T + 3.29e15iT^{2} \) |
| 59 | \( 1 + (-2.07e7 + 2.07e7i)T - 8.66e15iT^{2} \) |
| 61 | \( 1 + (-1.04e8 - 4.31e7i)T + (8.26e15 + 8.26e15i)T^{2} \) |
| 67 | \( 1 + 2.57e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + (4.35e7 + 1.05e8i)T + (-3.24e16 + 3.24e16i)T^{2} \) |
| 73 | \( 1 + (7.97e7 - 3.30e7i)T + (4.16e16 - 4.16e16i)T^{2} \) |
| 79 | \( 1 + (-2.35e8 + 5.68e8i)T + (-8.47e16 - 8.47e16i)T^{2} \) |
| 83 | \( 1 + (-3.74e8 - 3.74e8i)T + 1.86e17iT^{2} \) |
| 89 | \( 1 + 2.99e8iT - 3.50e17T^{2} \) |
| 97 | \( 1 + (-1.42e9 + 5.88e8i)T + (5.37e17 - 5.37e17i)T^{2} \) |
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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
−16.54088431447398849087052439392, −14.16826221220926738720212535841, −13.37874994486216576822695090981, −11.98970193121987959621307542889, −10.40200697372949490770844512487, −8.891333603726532612060776736236, −6.49228125889624494904357074590, −5.73952766291287761537396704165, −1.80931267183117146272053605145, −0.75906921107050307198530131172,
3.07763240689555227076185615302, 5.07001115712438080600222404076, 6.85498296305139880973523097987, 9.339688865555864467350466781443, 10.03799549000404706743113837988, 11.73211855479044075515189099103, 13.40577847641690094158006142096, 15.17253025587269110891978989890, 16.18204661673458038904486540056, 17.41645490869621593788340722519
