L(s) = 1 | + (−22.1 − 4.62i)2-s + (107. − 107. i)3-s + (469. + 204. i)4-s + (−103. + 1.39e3i)5-s + (−2.88e3 + 1.88e3i)6-s + (−1.03e3 − 1.03e3i)7-s + (−9.44e3 − 6.71e3i)8-s − 3.49e3i·9-s + (8.74e3 − 3.03e4i)10-s + 5.58e4i·11-s + (7.25e4 − 2.84e4i)12-s + (7.70e4 + 7.70e4i)13-s + (1.81e4 + 2.76e4i)14-s + (1.38e5 + 1.61e5i)15-s + (1.78e5 + 1.92e5i)16-s + (1.32e5 − 1.32e5i)17-s + ⋯ |
L(s) = 1 | + (−0.978 − 0.204i)2-s + (0.767 − 0.767i)3-s + (0.916 + 0.400i)4-s + (−0.0740 + 0.997i)5-s + (−0.907 + 0.594i)6-s + (−0.162 − 0.162i)7-s + (−0.815 − 0.579i)8-s − 0.177i·9-s + (0.276 − 0.961i)10-s + 1.15i·11-s + (1.01 − 0.395i)12-s + (0.748 + 0.748i)13-s + (0.125 + 0.192i)14-s + (0.708 + 0.821i)15-s + (0.679 + 0.733i)16-s + (0.384 − 0.384i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.862 - 0.506i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.862 - 0.506i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(5)\) |
\(\approx\) |
\(1.33015 + 0.361812i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.33015 + 0.361812i\) |
\(L(\frac{11}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (22.1 + 4.62i)T \) |
| 5 | \( 1 + (103. - 1.39e3i)T \) |
good | 3 | \( 1 + (-107. + 107. i)T - 1.96e4iT^{2} \) |
| 7 | \( 1 + (1.03e3 + 1.03e3i)T + 4.03e7iT^{2} \) |
| 11 | \( 1 - 5.58e4iT - 2.35e9T^{2} \) |
| 13 | \( 1 + (-7.70e4 - 7.70e4i)T + 1.06e10iT^{2} \) |
| 17 | \( 1 + (-1.32e5 + 1.32e5i)T - 1.18e11iT^{2} \) |
| 19 | \( 1 - 6.83e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + (-4.98e5 + 4.98e5i)T - 1.80e12iT^{2} \) |
| 29 | \( 1 - 6.68e6iT - 1.45e13T^{2} \) |
| 31 | \( 1 + 3.14e6iT - 2.64e13T^{2} \) |
| 37 | \( 1 + (-3.82e6 + 3.82e6i)T - 1.29e14iT^{2} \) |
| 41 | \( 1 + 2.79e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + (6.41e6 - 6.41e6i)T - 5.02e14iT^{2} \) |
| 47 | \( 1 + (-3.41e7 - 3.41e7i)T + 1.11e15iT^{2} \) |
| 53 | \( 1 + (-4.49e7 - 4.49e7i)T + 3.29e15iT^{2} \) |
| 59 | \( 1 + 1.33e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 4.19e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + (6.05e7 + 6.05e7i)T + 2.72e16iT^{2} \) |
| 71 | \( 1 + 2.54e8iT - 4.58e16T^{2} \) |
| 73 | \( 1 + (2.12e8 + 2.12e8i)T + 5.88e16iT^{2} \) |
| 79 | \( 1 - 3.42e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + (-9.07e7 + 9.07e7i)T - 1.86e17iT^{2} \) |
| 89 | \( 1 - 1.04e9iT - 3.50e17T^{2} \) |
| 97 | \( 1 + (1.03e8 - 1.03e8i)T - 7.60e17iT^{2} \) |
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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
−16.45382420763329632683039867219, −15.00706944067952868000415188990, −13.73029708015610114110800866506, −12.12562903930482437769621644777, −10.67509304444419553109282170819, −9.256228596644997450677156559670, −7.63209976404510025920742523113, −6.84164792095004914245778121925, −3.08063197643914094190402900079, −1.64178738882160584563438746766,
0.875623680217385421614739095514, 3.32192357130080472737241430022, 5.72034723387270327609252702618, 8.123638969143427193493346616480, 8.950568305549719625718879083512, 10.10356033321175128063568028135, 11.76109898864716696605861328683, 13.65826310067103902184587422718, 15.33735158101053258064348001061, 16.00902028076509878032963088337