| L(s) = 1 | + (28.7 + 28.7i)2-s + (39.4 + 95.1i)3-s + 1.13e3i·4-s + (710. − 294. i)5-s + (−1.60e3 + 3.86e3i)6-s + (−1.44e3 − 599. i)7-s + (−1.79e4 + 1.79e4i)8-s + (6.42e3 − 6.42e3i)9-s + (2.88e4 + 1.19e4i)10-s + (−6.51e3 + 1.57e4i)11-s + (−1.08e5 + 4.48e4i)12-s − 1.41e5i·13-s + (−2.43e4 − 5.88e4i)14-s + (5.60e4 + 5.60e4i)15-s − 4.48e5·16-s + (−2.13e5 + 2.69e5i)17-s + ⋯ |
| L(s) = 1 | + (1.26 + 1.26i)2-s + (0.280 + 0.678i)3-s + 2.22i·4-s + (0.508 − 0.210i)5-s + (−0.504 + 1.21i)6-s + (−0.227 − 0.0944i)7-s + (−1.54 + 1.54i)8-s + (0.326 − 0.326i)9-s + (0.912 + 0.378i)10-s + (−0.134 + 0.324i)11-s + (−1.50 + 0.623i)12-s − 1.37i·13-s + (−0.169 − 0.409i)14-s + (0.285 + 0.285i)15-s − 1.71·16-s + (−0.620 + 0.784i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.734 - 0.678i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.734 - 0.678i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.32347 + 3.38380i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.32347 + 3.38380i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (2.13e5 - 2.69e5i)T \) |
| good | 2 | \( 1 + (-28.7 - 28.7i)T + 512iT^{2} \) |
| 3 | \( 1 + (-39.4 - 95.1i)T + (-1.39e4 + 1.39e4i)T^{2} \) |
| 5 | \( 1 + (-710. + 294. i)T + (1.38e6 - 1.38e6i)T^{2} \) |
| 7 | \( 1 + (1.44e3 + 599. i)T + (2.85e7 + 2.85e7i)T^{2} \) |
| 11 | \( 1 + (6.51e3 - 1.57e4i)T + (-1.66e9 - 1.66e9i)T^{2} \) |
| 13 | \( 1 + 1.41e5iT - 1.06e10T^{2} \) |
| 19 | \( 1 + (-5.25e5 - 5.25e5i)T + 3.22e11iT^{2} \) |
| 23 | \( 1 + (-5.95e5 + 1.43e6i)T + (-1.27e12 - 1.27e12i)T^{2} \) |
| 29 | \( 1 + (-5.30e6 + 2.19e6i)T + (1.02e13 - 1.02e13i)T^{2} \) |
| 31 | \( 1 + (1.75e6 + 4.24e6i)T + (-1.86e13 + 1.86e13i)T^{2} \) |
| 37 | \( 1 + (-2.09e6 - 5.06e6i)T + (-9.18e13 + 9.18e13i)T^{2} \) |
| 41 | \( 1 + (1.67e7 + 6.93e6i)T + (2.31e14 + 2.31e14i)T^{2} \) |
| 43 | \( 1 + (1.97e7 - 1.97e7i)T - 5.02e14iT^{2} \) |
| 47 | \( 1 + 2.65e7iT - 1.11e15T^{2} \) |
| 53 | \( 1 + (5.36e7 + 5.36e7i)T + 3.29e15iT^{2} \) |
| 59 | \( 1 + (3.36e7 - 3.36e7i)T - 8.66e15iT^{2} \) |
| 61 | \( 1 + (1.29e8 + 5.36e7i)T + (8.26e15 + 8.26e15i)T^{2} \) |
| 67 | \( 1 - 7.10e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + (-2.73e7 - 6.59e7i)T + (-3.24e16 + 3.24e16i)T^{2} \) |
| 73 | \( 1 + (-3.14e8 + 1.30e8i)T + (4.16e16 - 4.16e16i)T^{2} \) |
| 79 | \( 1 + (-1.22e8 + 2.96e8i)T + (-8.47e16 - 8.47e16i)T^{2} \) |
| 83 | \( 1 + (-2.30e8 - 2.30e8i)T + 1.86e17iT^{2} \) |
| 89 | \( 1 + 2.75e8iT - 3.50e17T^{2} \) |
| 97 | \( 1 + (6.15e8 - 2.54e8i)T + (5.37e17 - 5.37e17i)T^{2} \) |
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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
−16.71617933426227509272936092588, −15.53632242985114107456908392865, −14.86058692149059556693356077877, −13.45114293051416602102810419112, −12.48766608058291506485576190478, −10.00558931776632439081028859119, −8.128294319433107329267941937045, −6.38051904635808172689749368826, −4.94110456254015760655248879100, −3.44408719092744647939695452502,
1.51940259389029848096174671848, 2.84646140694464102652509414790, 4.85717147878918204947394411210, 6.73126950917738810475387273344, 9.496911495992985651972753326776, 11.06396234804011880757678374644, 12.25935569036866836202919679533, 13.74581729499978514242084335113, 13.82858141523224021952901846627, 15.81626653803486754452288729614
