L(s) = 1 | + (−2.23 − 2.23i)2-s + (−45.0 + 18.6i)3-s − 502. i·4-s + (265. + 640. i)5-s + (142. + 58.9i)6-s + (−537. + 1.29e3i)7-s + (−2.26e3 + 2.26e3i)8-s + (−1.22e4 + 1.22e4i)9-s + (838. − 2.02e3i)10-s + (−6.54e4 − 2.71e4i)11-s + (9.36e3 + 2.25e4i)12-s + 1.80e5i·13-s + (4.09e3 − 1.69e3i)14-s + (−2.38e4 − 2.38e4i)15-s − 2.46e5·16-s + (4.12e4 + 3.41e5i)17-s + ⋯ |
L(s) = 1 | + (−0.0987 − 0.0987i)2-s + (−0.320 + 0.132i)3-s − 0.980i·4-s + (0.189 + 0.458i)5-s + (0.0448 + 0.0185i)6-s + (−0.0845 + 0.204i)7-s + (−0.195 + 0.195i)8-s + (−0.621 + 0.621i)9-s + (0.0265 − 0.0639i)10-s + (−1.34 − 0.558i)11-s + (0.130 + 0.314i)12-s + 1.75i·13-s + (0.0285 − 0.0118i)14-s + (−0.121 − 0.121i)15-s − 0.941·16-s + (0.119 + 0.992i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.757 - 0.653i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.757 - 0.653i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(5)\) |
\(\approx\) |
\(0.123713 + 0.332707i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.123713 + 0.332707i\) |
\(L(\frac{11}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 + (-4.12e4 - 3.41e5i)T \) |
good | 2 | \( 1 + (2.23 + 2.23i)T + 512iT^{2} \) |
| 3 | \( 1 + (45.0 - 18.6i)T + (1.39e4 - 1.39e4i)T^{2} \) |
| 5 | \( 1 + (-265. - 640. i)T + (-1.38e6 + 1.38e6i)T^{2} \) |
| 7 | \( 1 + (537. - 1.29e3i)T + (-2.85e7 - 2.85e7i)T^{2} \) |
| 11 | \( 1 + (6.54e4 + 2.71e4i)T + (1.66e9 + 1.66e9i)T^{2} \) |
| 13 | \( 1 - 1.80e5iT - 1.06e10T^{2} \) |
| 19 | \( 1 + (5.99e5 + 5.99e5i)T + 3.22e11iT^{2} \) |
| 23 | \( 1 + (-1.45e6 - 6.03e5i)T + (1.27e12 + 1.27e12i)T^{2} \) |
| 29 | \( 1 + (1.18e6 + 2.86e6i)T + (-1.02e13 + 1.02e13i)T^{2} \) |
| 31 | \( 1 + (3.10e6 - 1.28e6i)T + (1.86e13 - 1.86e13i)T^{2} \) |
| 37 | \( 1 + (1.75e7 - 7.27e6i)T + (9.18e13 - 9.18e13i)T^{2} \) |
| 41 | \( 1 + (-3.30e6 + 7.98e6i)T + (-2.31e14 - 2.31e14i)T^{2} \) |
| 43 | \( 1 + (7.27e6 - 7.27e6i)T - 5.02e14iT^{2} \) |
| 47 | \( 1 + 5.71e6iT - 1.11e15T^{2} \) |
| 53 | \( 1 + (5.32e7 + 5.32e7i)T + 3.29e15iT^{2} \) |
| 59 | \( 1 + (-1.00e8 + 1.00e8i)T - 8.66e15iT^{2} \) |
| 61 | \( 1 + (-4.03e4 + 9.74e4i)T + (-8.26e15 - 8.26e15i)T^{2} \) |
| 67 | \( 1 + 1.57e6T + 2.72e16T^{2} \) |
| 71 | \( 1 + (3.00e8 - 1.24e8i)T + (3.24e16 - 3.24e16i)T^{2} \) |
| 73 | \( 1 + (7.03e6 + 1.69e7i)T + (-4.16e16 + 4.16e16i)T^{2} \) |
| 79 | \( 1 + (-1.35e8 - 5.63e7i)T + (8.47e16 + 8.47e16i)T^{2} \) |
| 83 | \( 1 + (-2.62e8 - 2.62e8i)T + 1.86e17iT^{2} \) |
| 89 | \( 1 + 3.54e8iT - 3.50e17T^{2} \) |
| 97 | \( 1 + (-5.89e8 - 1.42e9i)T + (-5.37e17 + 5.37e17i)T^{2} \) |
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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
−17.26536101662279209857742112187, −15.86956359890694586322282415804, −14.58853375613273066259772966230, −13.41151261718665122021380225132, −11.25551737739491127463788152790, −10.50065536384180756817483554377, −8.795148805085886710188682999648, −6.49685966584458523164411826248, −5.08163236425657428090584378628, −2.24895818913489888925930773830,
0.17606778700386965858400396365, 3.05104156539369474832351707188, 5.32049165015847802630806432744, 7.37906705677569696479528803992, 8.736096537068800694951451137418, 10.62332480162201495948651070456, 12.42773988547731735704293022918, 13.03686800746006500626631355412, 15.08734932317466057100733510456, 16.48001728986359881832365377094