| L(s) = 1 | + (−38.9 − 16.1i)2-s + (−71.3 + 14.1i)3-s + (−1.64e3 − 1.64e3i)4-s + (−4.25e3 + 6.37e3i)5-s + (3.00e3 + 597. i)6-s + (−7.75e4 − 1.16e5i)7-s + (1.03e5 + 2.49e5i)8-s + (−4.86e5 + 2.01e5i)9-s + (2.68e5 − 1.79e5i)10-s + (3.65e5 − 1.83e6i)11-s + (1.40e5 + 9.38e4i)12-s + (−2.94e6 + 2.94e6i)13-s + (1.14e6 + 5.76e6i)14-s + (2.13e5 − 5.15e5i)15-s − 1.87e6i·16-s + (2.40e7 + 2.29e6i)17-s + ⋯ |
| L(s) = 1 | + (−0.608 − 0.251i)2-s + (−0.0978 + 0.0194i)3-s + (−0.400 − 0.400i)4-s + (−0.272 + 0.407i)5-s + (0.0644 + 0.0128i)6-s + (−0.658 − 0.986i)7-s + (0.394 + 0.952i)8-s + (−0.914 + 0.378i)9-s + (0.268 − 0.179i)10-s + (0.206 − 1.03i)11-s + (0.0470 + 0.0314i)12-s + (−0.610 + 0.610i)13-s + (0.152 + 0.765i)14-s + (0.0187 − 0.0452i)15-s − 0.111i·16-s + (0.995 + 0.0949i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.875 - 0.483i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (0.875 - 0.483i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{13}{2})\) |
\(\approx\) |
\(0.641632 + 0.165239i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.641632 + 0.165239i\) |
| \(L(7)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (-2.40e7 - 2.29e6i)T \) |
| good | 2 | \( 1 + (38.9 + 16.1i)T + (2.89e3 + 2.89e3i)T^{2} \) |
| 3 | \( 1 + (71.3 - 14.1i)T + (4.90e5 - 2.03e5i)T^{2} \) |
| 5 | \( 1 + (4.25e3 - 6.37e3i)T + (-9.34e7 - 2.25e8i)T^{2} \) |
| 7 | \( 1 + (7.75e4 + 1.16e5i)T + (-5.29e9 + 1.27e10i)T^{2} \) |
| 11 | \( 1 + (-3.65e5 + 1.83e6i)T + (-2.89e12 - 1.20e12i)T^{2} \) |
| 13 | \( 1 + (2.94e6 - 2.94e6i)T - 2.32e13iT^{2} \) |
| 19 | \( 1 + (-6.13e7 - 2.54e7i)T + (1.56e15 + 1.56e15i)T^{2} \) |
| 23 | \( 1 + (-1.90e8 - 3.78e7i)T + (2.02e16 + 8.38e15i)T^{2} \) |
| 29 | \( 1 + (6.66e8 + 4.45e8i)T + (1.35e17 + 3.26e17i)T^{2} \) |
| 31 | \( 1 + (-1.22e8 - 6.15e8i)T + (-7.27e17 + 3.01e17i)T^{2} \) |
| 37 | \( 1 + (3.97e9 - 7.91e8i)T + (6.08e18 - 2.51e18i)T^{2} \) |
| 41 | \( 1 + (2.95e8 + 4.41e8i)T + (-8.63e18 + 2.08e19i)T^{2} \) |
| 43 | \( 1 + (-1.10e10 + 4.55e9i)T + (2.82e19 - 2.82e19i)T^{2} \) |
| 47 | \( 1 + (1.35e10 - 1.35e10i)T - 1.16e20iT^{2} \) |
| 53 | \( 1 + (-2.15e10 - 8.94e9i)T + (3.47e20 + 3.47e20i)T^{2} \) |
| 59 | \( 1 + (-1.89e10 - 4.57e10i)T + (-1.25e21 + 1.25e21i)T^{2} \) |
| 61 | \( 1 + (-1.69e10 + 1.13e10i)T + (1.01e21 - 2.45e21i)T^{2} \) |
| 67 | \( 1 + 4.75e10iT - 8.18e21T^{2} \) |
| 71 | \( 1 + (1.32e11 - 2.64e10i)T + (1.51e22 - 6.27e21i)T^{2} \) |
| 73 | \( 1 + (7.20e10 - 1.07e11i)T + (-8.76e21 - 2.11e22i)T^{2} \) |
| 79 | \( 1 + (-3.87e10 + 1.94e11i)T + (-5.45e22 - 2.26e22i)T^{2} \) |
| 83 | \( 1 + (4.33e10 - 1.04e11i)T + (-7.55e22 - 7.55e22i)T^{2} \) |
| 89 | \( 1 + (-3.48e11 - 3.48e11i)T + 2.46e23iT^{2} \) |
| 97 | \( 1 + (-5.10e11 - 3.41e11i)T + (2.65e23 + 6.41e23i)T^{2} \) |
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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
−16.49280571769986313552222741457, −14.45497095421228139412542143700, −13.66135209115466833556108580607, −11.50539579588188458228612889975, −10.42876584649199740185967740835, −9.119420512932718331753724211301, −7.46203905991324093837843557231, −5.48944004392914405446977854197, −3.33228298787249016672599764339, −0.941205007530869621946096188803,
0.45806185366473098468320036527, 3.12295966946721482223270878309, 5.24490243868970386675011371149, 7.26733654769083433484610128922, 8.788855085103616230803703891666, 9.697925443321669071444964842063, 12.00579605624306955978687547682, 12.81245558819885847821364461309, 14.76520054837305437323923748003, 16.09630524542749421834849671538
