| L(s) = 1 | + (6.30 + 6.30i)2-s + (12.9 + 31.3i)3-s − 432. i·4-s + (−32.2 + 13.3i)5-s + (−115. + 279. i)6-s + (−7.88e3 − 3.26e3i)7-s + (5.95e3 − 5.95e3i)8-s + (1.31e4 − 1.31e4i)9-s + (−287. − 119. i)10-s + (317. − 765. i)11-s + (1.35e4 − 5.61e3i)12-s − 1.47e5i·13-s + (−2.91e4 − 7.03e4i)14-s + (−836. − 836. i)15-s − 1.46e5·16-s + (3.38e5 − 6.21e4i)17-s + ⋯ |
| L(s) = 1 | + (0.278 + 0.278i)2-s + (0.0925 + 0.223i)3-s − 0.844i·4-s + (−0.0230 + 0.00954i)5-s + (−0.0364 + 0.0880i)6-s + (−1.24 − 0.514i)7-s + (0.514 − 0.514i)8-s + (0.665 − 0.665i)9-s + (−0.00908 − 0.00376i)10-s + (0.00653 − 0.0157i)11-s + (0.188 − 0.0781i)12-s − 1.43i·13-s + (−0.202 − 0.489i)14-s + (−0.00426 − 0.00426i)15-s − 0.558·16-s + (0.983 − 0.180i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.105 + 0.994i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.105 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.17241 - 1.05494i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.17241 - 1.05494i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (-3.38e5 + 6.21e4i)T \) |
| good | 2 | \( 1 + (-6.30 - 6.30i)T + 512iT^{2} \) |
| 3 | \( 1 + (-12.9 - 31.3i)T + (-1.39e4 + 1.39e4i)T^{2} \) |
| 5 | \( 1 + (32.2 - 13.3i)T + (1.38e6 - 1.38e6i)T^{2} \) |
| 7 | \( 1 + (7.88e3 + 3.26e3i)T + (2.85e7 + 2.85e7i)T^{2} \) |
| 11 | \( 1 + (-317. + 765. i)T + (-1.66e9 - 1.66e9i)T^{2} \) |
| 13 | \( 1 + 1.47e5iT - 1.06e10T^{2} \) |
| 19 | \( 1 + (2.72e5 + 2.72e5i)T + 3.22e11iT^{2} \) |
| 23 | \( 1 + (7.57e5 - 1.82e6i)T + (-1.27e12 - 1.27e12i)T^{2} \) |
| 29 | \( 1 + (-2.43e6 + 1.00e6i)T + (1.02e13 - 1.02e13i)T^{2} \) |
| 31 | \( 1 + (-2.45e6 - 5.92e6i)T + (-1.86e13 + 1.86e13i)T^{2} \) |
| 37 | \( 1 + (7.24e6 + 1.74e7i)T + (-9.18e13 + 9.18e13i)T^{2} \) |
| 41 | \( 1 + (-8.61e6 - 3.56e6i)T + (2.31e14 + 2.31e14i)T^{2} \) |
| 43 | \( 1 + (-1.21e7 + 1.21e7i)T - 5.02e14iT^{2} \) |
| 47 | \( 1 + 1.46e7iT - 1.11e15T^{2} \) |
| 53 | \( 1 + (-7.04e7 - 7.04e7i)T + 3.29e15iT^{2} \) |
| 59 | \( 1 + (-5.20e7 + 5.20e7i)T - 8.66e15iT^{2} \) |
| 61 | \( 1 + (5.94e5 + 2.46e5i)T + (8.26e15 + 8.26e15i)T^{2} \) |
| 67 | \( 1 - 2.31e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + (1.67e7 + 4.04e7i)T + (-3.24e16 + 3.24e16i)T^{2} \) |
| 73 | \( 1 + (-1.93e8 + 8.02e7i)T + (4.16e16 - 4.16e16i)T^{2} \) |
| 79 | \( 1 + (2.74e7 - 6.63e7i)T + (-8.47e16 - 8.47e16i)T^{2} \) |
| 83 | \( 1 + (2.80e8 + 2.80e8i)T + 1.86e17iT^{2} \) |
| 89 | \( 1 - 2.63e8iT - 3.50e17T^{2} \) |
| 97 | \( 1 + (-2.77e8 + 1.14e8i)T + (5.37e17 - 5.37e17i)T^{2} \) |
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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
−15.96094902618458662565345421514, −15.31926185876756180230178287407, −13.79337583449027475187518924130, −12.61267533324911186663154863194, −10.40855777097154001863683250777, −9.597779795224046791728995918558, −7.14639530708357975364407876391, −5.67937270363443465490683736723, −3.62875759127112893866073122643, −0.72484054985232978524984610039,
2.36426090839556735798163993281, 4.15118257183561840392759898078, 6.59864523601653195931608281761, 8.257721763541216481669495212150, 9.987261363417058333662319639424, 11.94739850368355783421608232666, 12.81324673211265977008786609361, 14.02390066777514475227261256693, 16.04828190552680401332372948132, 16.73497891397183008632654160167
