| L(s) = 1 | + (627. + 362. i)2-s + (4.74e4 + 3.51e4i)3-s + (2.62e5 + 4.54e5i)4-s + (5.33e6 − 3.07e6i)5-s + (1.70e7 + 3.92e7i)6-s + (−2.63e8 + 4.56e8i)7-s + 3.79e8i·8-s + (1.02e9 + 3.33e9i)9-s + 4.45e9·10-s + (−3.42e10 − 1.98e10i)11-s + (−3.49e9 + 3.07e10i)12-s + (−7.57e10 − 1.31e11i)13-s + (−3.30e11 + 1.90e11i)14-s + (3.61e11 + 4.09e10i)15-s + (−1.37e11 + 2.38e11i)16-s + 1.43e12i·17-s + ⋯ |
| L(s) = 1 | + (0.612 + 0.353i)2-s + (0.804 + 0.594i)3-s + (0.249 + 0.433i)4-s + (0.545 − 0.315i)5-s + (0.282 + 0.648i)6-s + (−0.932 + 1.61i)7-s + 0.353i·8-s + (0.293 + 0.956i)9-s + 0.445·10-s + (−1.32 − 0.763i)11-s + (−0.0563 + 0.496i)12-s + (−0.549 − 0.951i)13-s + (−1.14 + 0.659i)14-s + (0.626 + 0.0710i)15-s + (−0.125 + 0.216i)16-s + 0.712i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.122i)\, \overline{\Lambda}(21-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+10) \, L(s)\cr =\mathstrut & (-0.992 + 0.122i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{21}{2})\) |
\(\approx\) |
\(2.191190639\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.191190639\) |
| \(L(11)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-627. - 362. i)T \) |
| 3 | \( 1 + (-4.74e4 - 3.51e4i)T \) |
| good | 5 | \( 1 + (-5.33e6 + 3.07e6i)T + (4.76e13 - 8.25e13i)T^{2} \) |
| 7 | \( 1 + (2.63e8 - 4.56e8i)T + (-3.98e16 - 6.91e16i)T^{2} \) |
| 11 | \( 1 + (3.42e10 + 1.98e10i)T + (3.36e20 + 5.82e20i)T^{2} \) |
| 13 | \( 1 + (7.57e10 + 1.31e11i)T + (-9.50e21 + 1.64e22i)T^{2} \) |
| 17 | \( 1 - 1.43e12iT - 4.06e24T^{2} \) |
| 19 | \( 1 + 5.01e12T + 3.75e25T^{2} \) |
| 23 | \( 1 + (-3.78e13 + 2.18e13i)T + (8.58e26 - 1.48e27i)T^{2} \) |
| 29 | \( 1 + (-3.69e14 - 2.13e14i)T + (8.84e28 + 1.53e29i)T^{2} \) |
| 31 | \( 1 + (2.91e14 + 5.05e14i)T + (-3.35e29 + 5.81e29i)T^{2} \) |
| 37 | \( 1 - 1.98e15T + 2.31e31T^{2} \) |
| 41 | \( 1 + (4.29e15 - 2.47e15i)T + (9.00e31 - 1.56e32i)T^{2} \) |
| 43 | \( 1 + (1.01e16 - 1.75e16i)T + (-2.33e32 - 4.04e32i)T^{2} \) |
| 47 | \( 1 + (-5.23e16 - 3.02e16i)T + (1.38e33 + 2.39e33i)T^{2} \) |
| 53 | \( 1 + 1.27e17iT - 3.05e34T^{2} \) |
| 59 | \( 1 + (7.01e17 - 4.05e17i)T + (1.30e35 - 2.26e35i)T^{2} \) |
| 61 | \( 1 + (6.21e17 - 1.07e18i)T + (-2.54e35 - 4.40e35i)T^{2} \) |
| 67 | \( 1 + (5.17e17 + 8.95e17i)T + (-1.66e36 + 2.87e36i)T^{2} \) |
| 71 | \( 1 + 1.04e18iT - 1.05e37T^{2} \) |
| 73 | \( 1 - 6.74e18T + 1.84e37T^{2} \) |
| 79 | \( 1 + (5.28e18 - 9.15e18i)T + (-4.48e37 - 7.76e37i)T^{2} \) |
| 83 | \( 1 + (-6.27e18 - 3.62e18i)T + (1.20e38 + 2.08e38i)T^{2} \) |
| 89 | \( 1 + 4.67e18iT - 9.72e38T^{2} \) |
| 97 | \( 1 + (1.18e19 - 2.05e19i)T + (-2.71e39 - 4.70e39i)T^{2} \) |
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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
−14.96200986907497771625812642120, −13.25525668339529334655818344180, −12.67166896484559816344117741829, −10.52189490951210320855629637610, −9.123871890892275915542943684790, −8.106515574188538191642575186947, −5.96585110616383088544320302832, −5.06384432877106978350455426051, −3.09462707201677575707374010580, −2.42857741208961668346614562738,
0.38192113095450857464989769035, 1.98939036192928203651407841243, 3.07818290414988392309300834327, 4.48701360439622751073313176174, 6.62249142030053145104790683932, 7.40652800023788984977267224373, 9.593853855452266394413119533143, 10.49576037976291390025338233540, 12.48430981187107493392472082328, 13.49137061520393427295957573888
