| L(s) = 1 | + (−29.1 − 29.1i)2-s + (−162. + 67.1i)3-s + 1.19e3i·4-s + (−267. − 646. i)5-s + (6.68e3 + 2.76e3i)6-s + (−3.38e3 + 8.17e3i)7-s + (1.98e4 − 1.98e4i)8-s + (7.83e3 − 7.83e3i)9-s + (−1.10e4 + 2.66e4i)10-s + (−8.36e4 − 3.46e4i)11-s + (−7.99e4 − 1.92e5i)12-s + 2.62e4i·13-s + (3.37e5 − 1.39e5i)14-s + (8.68e4 + 8.68e4i)15-s − 5.46e5·16-s + (2.09e5 − 2.73e5i)17-s + ⋯ |
| L(s) = 1 | + (−1.28 − 1.28i)2-s + (−1.15 + 0.478i)3-s + 2.32i·4-s + (−0.191 − 0.462i)5-s + (2.10 + 0.872i)6-s + (−0.533 + 1.28i)7-s + (1.70 − 1.70i)8-s + (0.397 − 0.397i)9-s + (−0.349 + 0.843i)10-s + (−1.72 − 0.713i)11-s + (−1.11 − 2.68i)12-s + 0.254i·13-s + (2.34 − 0.972i)14-s + (0.442 + 0.442i)15-s − 2.08·16-s + (0.608 − 0.793i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0847 + 0.996i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.0847 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.203246 - 0.186684i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.203246 - 0.186684i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (-2.09e5 + 2.73e5i)T \) |
| good | 2 | \( 1 + (29.1 + 29.1i)T + 512iT^{2} \) |
| 3 | \( 1 + (162. - 67.1i)T + (1.39e4 - 1.39e4i)T^{2} \) |
| 5 | \( 1 + (267. + 646. i)T + (-1.38e6 + 1.38e6i)T^{2} \) |
| 7 | \( 1 + (3.38e3 - 8.17e3i)T + (-2.85e7 - 2.85e7i)T^{2} \) |
| 11 | \( 1 + (8.36e4 + 3.46e4i)T + (1.66e9 + 1.66e9i)T^{2} \) |
| 13 | \( 1 - 2.62e4iT - 1.06e10T^{2} \) |
| 19 | \( 1 + (-2.68e5 - 2.68e5i)T + 3.22e11iT^{2} \) |
| 23 | \( 1 + (1.84e5 + 7.62e4i)T + (1.27e12 + 1.27e12i)T^{2} \) |
| 29 | \( 1 + (-6.30e5 - 1.52e6i)T + (-1.02e13 + 1.02e13i)T^{2} \) |
| 31 | \( 1 + (5.69e6 - 2.35e6i)T + (1.86e13 - 1.86e13i)T^{2} \) |
| 37 | \( 1 + (-1.85e7 + 7.67e6i)T + (9.18e13 - 9.18e13i)T^{2} \) |
| 41 | \( 1 + (9.72e6 - 2.34e7i)T + (-2.31e14 - 2.31e14i)T^{2} \) |
| 43 | \( 1 + (3.15e5 - 3.15e5i)T - 5.02e14iT^{2} \) |
| 47 | \( 1 + 1.95e7iT - 1.11e15T^{2} \) |
| 53 | \( 1 + (-4.24e7 - 4.24e7i)T + 3.29e15iT^{2} \) |
| 59 | \( 1 + (-7.36e6 + 7.36e6i)T - 8.66e15iT^{2} \) |
| 61 | \( 1 + (-5.62e7 + 1.35e8i)T + (-8.26e15 - 8.26e15i)T^{2} \) |
| 67 | \( 1 - 1.11e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + (-8.95e7 + 3.71e7i)T + (3.24e16 - 3.24e16i)T^{2} \) |
| 73 | \( 1 + (2.39e6 + 5.78e6i)T + (-4.16e16 + 4.16e16i)T^{2} \) |
| 79 | \( 1 + (3.10e8 + 1.28e8i)T + (8.47e16 + 8.47e16i)T^{2} \) |
| 83 | \( 1 + (-3.37e8 - 3.37e8i)T + 1.86e17iT^{2} \) |
| 89 | \( 1 + 1.92e8iT - 3.50e17T^{2} \) |
| 97 | \( 1 + (8.11e7 + 1.96e8i)T + (-5.37e17 + 5.37e17i)T^{2} \) |
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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
−16.45705945973178052010115863588, −16.10171121662632285225799444491, −12.76579466141564769652935572752, −11.84880711075833840037517954954, −10.79152265604319128639609043374, −9.590472732455740671509648854459, −8.205904223028275574262167501877, −5.39788856511890001579296866681, −2.78915489548303381363753587443, −0.41873242383969445587448356887,
0.69763872320683976063044698510, 5.51014439154985838483779131298, 6.93481772813025270694997846598, 7.70101949239009114991735763413, 10.03934023261472474995018859333, 10.84620677284642148062058932543, 13.09462709623194171831711990742, 14.97554148058536176978708101597, 16.21864878779332363451238087529, 17.11451174398691264436202238861
