| L(s) = 1 | + (31.9 + 31.9i)2-s + (53.8 + 130. i)3-s − 1.67i·4-s + (1.19e4 − 4.95e3i)5-s + (−2.43e3 + 5.88e3i)6-s + (−6.03e4 − 2.49e4i)7-s + (6.55e4 − 6.55e4i)8-s + (1.11e5 − 1.11e5i)9-s + (5.40e5 + 2.24e5i)10-s + (8.32e4 − 2.00e5i)11-s + (217. − 90.1i)12-s + 9.55e5i·13-s + (−1.13e6 − 2.72e6i)14-s + (1.28e6 + 1.28e6i)15-s + 4.19e6·16-s + (5.50e5 + 5.82e6i)17-s + ⋯ |
| L(s) = 1 | + (0.706 + 0.706i)2-s + (0.127 + 0.309i)3-s − 0.000817i·4-s + (1.71 − 0.708i)5-s + (−0.127 + 0.308i)6-s + (−1.35 − 0.562i)7-s + (0.707 − 0.707i)8-s + (0.628 − 0.628i)9-s + (1.71 + 0.708i)10-s + (0.155 − 0.376i)11-s + (0.000252 − 0.000104i)12-s + 0.713i·13-s + (−0.561 − 1.35i)14-s + (0.437 + 0.437i)15-s + 0.999·16-s + (0.0940 + 0.995i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0182i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0182i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(3.31843 + 0.0302530i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.31843 + 0.0302530i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (-5.50e5 - 5.82e6i)T \) |
| good | 2 | \( 1 + (-31.9 - 31.9i)T + 2.04e3iT^{2} \) |
| 3 | \( 1 + (-53.8 - 130. i)T + (-1.25e5 + 1.25e5i)T^{2} \) |
| 5 | \( 1 + (-1.19e4 + 4.95e3i)T + (3.45e7 - 3.45e7i)T^{2} \) |
| 7 | \( 1 + (6.03e4 + 2.49e4i)T + (1.39e9 + 1.39e9i)T^{2} \) |
| 11 | \( 1 + (-8.32e4 + 2.00e5i)T + (-2.01e11 - 2.01e11i)T^{2} \) |
| 13 | \( 1 - 9.55e5iT - 1.79e12T^{2} \) |
| 19 | \( 1 + (-2.94e6 - 2.94e6i)T + 1.16e14iT^{2} \) |
| 23 | \( 1 + (7.37e6 - 1.78e7i)T + (-6.73e14 - 6.73e14i)T^{2} \) |
| 29 | \( 1 + (9.85e7 - 4.08e7i)T + (8.62e15 - 8.62e15i)T^{2} \) |
| 31 | \( 1 + (1.09e8 + 2.64e8i)T + (-1.79e16 + 1.79e16i)T^{2} \) |
| 37 | \( 1 + (-2.29e8 - 5.54e8i)T + (-1.25e17 + 1.25e17i)T^{2} \) |
| 41 | \( 1 + (-3.10e8 - 1.28e8i)T + (3.89e17 + 3.89e17i)T^{2} \) |
| 43 | \( 1 + (7.43e8 - 7.43e8i)T - 9.29e17iT^{2} \) |
| 47 | \( 1 + 1.07e9iT - 2.47e18T^{2} \) |
| 53 | \( 1 + (-5.38e8 - 5.38e8i)T + 9.26e18iT^{2} \) |
| 59 | \( 1 + (3.23e9 - 3.23e9i)T - 3.01e19iT^{2} \) |
| 61 | \( 1 + (-9.06e9 - 3.75e9i)T + (3.07e19 + 3.07e19i)T^{2} \) |
| 67 | \( 1 - 6.61e9T + 1.22e20T^{2} \) |
| 71 | \( 1 + (-2.15e9 - 5.20e9i)T + (-1.63e20 + 1.63e20i)T^{2} \) |
| 73 | \( 1 + (1.80e10 - 7.45e9i)T + (2.21e20 - 2.21e20i)T^{2} \) |
| 79 | \( 1 + (5.15e9 - 1.24e10i)T + (-5.28e20 - 5.28e20i)T^{2} \) |
| 83 | \( 1 + (9.66e9 + 9.66e9i)T + 1.28e21iT^{2} \) |
| 89 | \( 1 - 3.87e10iT - 2.77e21T^{2} \) |
| 97 | \( 1 + (7.28e9 - 3.01e9i)T + (5.05e21 - 5.05e21i)T^{2} \) |
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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
−16.29223043598102289586064821168, −14.76343157574544896745117345880, −13.45262780647558933829990946047, −12.95820680970414291589292048655, −10.02726300282386919488461853578, −9.463992661088266701693312645656, −6.63332816072464959938983421574, −5.76722270551547522315118797729, −3.97599891344395452064063638531, −1.30543385004599056396912991605,
2.05356968366239413028249451854, 3.00792385426195900480867337762, 5.43393185151608771477903773453, 7.00590684880500038399900973001, 9.504775138751283105576341851830, 10.60510357632819771698565995355, 12.60341627295550495516222991116, 13.25490150947852914292521539903, 14.27645222925855132333242563464, 16.21144514505767012689899125925
