| L(s) = 1 | + (−47.6 − 47.6i)2-s + (95.9 + 231. i)3-s + 2.48e3i·4-s + (−1.07e4 + 4.43e3i)5-s + (6.46e3 − 1.56e4i)6-s + (3.61e4 + 1.49e4i)7-s + (2.08e4 − 2.08e4i)8-s + (8.07e4 − 8.07e4i)9-s + (7.21e5 + 2.98e5i)10-s + (−1.38e5 + 3.33e5i)11-s + (−5.76e5 + 2.38e5i)12-s − 1.41e6i·13-s + (−1.00e6 − 2.43e6i)14-s + (−2.05e6 − 2.05e6i)15-s + 3.10e6·16-s + (−5.59e6 − 1.72e6i)17-s + ⋯ |
| L(s) = 1 | + (−1.05 − 1.05i)2-s + (0.228 + 0.550i)3-s + 1.21i·4-s + (−1.53 + 0.635i)5-s + (0.339 − 0.819i)6-s + (0.813 + 0.336i)7-s + (0.225 − 0.225i)8-s + (0.455 − 0.455i)9-s + (2.28 + 0.945i)10-s + (−0.258 + 0.624i)11-s + (−0.668 + 0.276i)12-s − 1.05i·13-s + (−0.501 − 1.21i)14-s + (−0.699 − 0.699i)15-s + 0.740·16-s + (−0.955 − 0.294i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.365 + 0.930i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.365 + 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.332710 - 0.488325i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.332710 - 0.488325i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (5.59e6 + 1.72e6i)T \) |
| good | 2 | \( 1 + (47.6 + 47.6i)T + 2.04e3iT^{2} \) |
| 3 | \( 1 + (-95.9 - 231. i)T + (-1.25e5 + 1.25e5i)T^{2} \) |
| 5 | \( 1 + (1.07e4 - 4.43e3i)T + (3.45e7 - 3.45e7i)T^{2} \) |
| 7 | \( 1 + (-3.61e4 - 1.49e4i)T + (1.39e9 + 1.39e9i)T^{2} \) |
| 11 | \( 1 + (1.38e5 - 3.33e5i)T + (-2.01e11 - 2.01e11i)T^{2} \) |
| 13 | \( 1 + 1.41e6iT - 1.79e12T^{2} \) |
| 19 | \( 1 + (5.45e6 + 5.45e6i)T + 1.16e14iT^{2} \) |
| 23 | \( 1 + (1.54e6 - 3.73e6i)T + (-6.73e14 - 6.73e14i)T^{2} \) |
| 29 | \( 1 + (-1.55e8 + 6.46e7i)T + (8.62e15 - 8.62e15i)T^{2} \) |
| 31 | \( 1 + (7.27e7 + 1.75e8i)T + (-1.79e16 + 1.79e16i)T^{2} \) |
| 37 | \( 1 + (3.66e7 + 8.84e7i)T + (-1.25e17 + 1.25e17i)T^{2} \) |
| 41 | \( 1 + (-6.26e8 - 2.59e8i)T + (3.89e17 + 3.89e17i)T^{2} \) |
| 43 | \( 1 + (-1.04e9 + 1.04e9i)T - 9.29e17iT^{2} \) |
| 47 | \( 1 + 1.18e9iT - 2.47e18T^{2} \) |
| 53 | \( 1 + (-3.47e9 - 3.47e9i)T + 9.26e18iT^{2} \) |
| 59 | \( 1 + (2.68e9 - 2.68e9i)T - 3.01e19iT^{2} \) |
| 61 | \( 1 + (2.01e9 + 8.35e8i)T + (3.07e19 + 3.07e19i)T^{2} \) |
| 67 | \( 1 - 3.11e9T + 1.22e20T^{2} \) |
| 71 | \( 1 + (5.69e9 + 1.37e10i)T + (-1.63e20 + 1.63e20i)T^{2} \) |
| 73 | \( 1 + (-4.16e9 + 1.72e9i)T + (2.21e20 - 2.21e20i)T^{2} \) |
| 79 | \( 1 + (1.09e10 - 2.63e10i)T + (-5.28e20 - 5.28e20i)T^{2} \) |
| 83 | \( 1 + (3.93e10 + 3.93e10i)T + 1.28e21iT^{2} \) |
| 89 | \( 1 + 5.80e10iT - 2.77e21T^{2} \) |
| 97 | \( 1 + (3.84e10 - 1.59e10i)T + (5.05e21 - 5.05e21i)T^{2} \) |
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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
−15.53339404874086231862221990371, −15.02272870844246440366932366043, −12.35909863647812183693562107430, −11.28337798614192405091114372260, −10.33881366239054456582812550363, −8.758048875097825728365380756922, −7.56271056859751439600065804152, −4.22055056671141358357452213649, −2.64022725940762228054300704881, −0.42367098297912807457769793848,
1.13951589091515335017896155546, 4.43418933529000534813276572315, 6.91772665132791915493975578223, 8.009277526071362158008722310744, 8.662975792062277832519136936979, 10.93530908056454302723060272428, 12.50808150168305784220881440387, 14.33420484151914920732587794404, 15.83735219521981968937805570957, 16.41007167409040220145158553162
