| L(s) = 1 | − 724. i·2-s − 5.24e5·4-s − 1.15e7i·5-s − 3.17e8·7-s + 3.79e8i·8-s − 8.36e9·10-s − 3.19e10i·11-s + 1.06e11·13-s + 2.29e11i·14-s + 2.74e11·16-s + 1.49e12i·17-s + 5.79e12·19-s + 6.05e12i·20-s − 2.31e13·22-s + 6.13e13i·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.500·4-s − 1.18i·5-s − 1.12·7-s + 0.353i·8-s − 0.836·10-s − 1.23i·11-s + 0.771·13-s + 0.794i·14-s + 0.250·16-s + 0.743i·17-s + 0.944·19-s + 0.591i·20-s − 0.870·22-s + 1.48i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(21-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+10) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{21}{2})\) |
\(\approx\) |
\(1.855209870\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.855209870\) |
| \(L(11)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 724. iT \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 1.15e7iT - 9.53e13T^{2} \) |
| 7 | \( 1 + 3.17e8T + 7.97e16T^{2} \) |
| 11 | \( 1 + 3.19e10iT - 6.72e20T^{2} \) |
| 13 | \( 1 - 1.06e11T + 1.90e22T^{2} \) |
| 17 | \( 1 - 1.49e12iT - 4.06e24T^{2} \) |
| 19 | \( 1 - 5.79e12T + 3.75e25T^{2} \) |
| 23 | \( 1 - 6.13e13iT - 1.71e27T^{2} \) |
| 29 | \( 1 - 2.01e14iT - 1.76e29T^{2} \) |
| 31 | \( 1 - 1.51e15T + 6.71e29T^{2} \) |
| 37 | \( 1 - 4.25e15T + 2.31e31T^{2} \) |
| 41 | \( 1 + 1.13e16iT - 1.80e32T^{2} \) |
| 43 | \( 1 + 3.42e16T + 4.67e32T^{2} \) |
| 47 | \( 1 - 3.39e16iT - 2.76e33T^{2} \) |
| 53 | \( 1 - 2.16e17iT - 3.05e34T^{2} \) |
| 59 | \( 1 - 9.09e16iT - 2.61e35T^{2} \) |
| 61 | \( 1 - 2.23e17T + 5.08e35T^{2} \) |
| 67 | \( 1 - 2.35e18T + 3.32e36T^{2} \) |
| 71 | \( 1 - 3.01e18iT - 1.05e37T^{2} \) |
| 73 | \( 1 - 3.66e18T + 1.84e37T^{2} \) |
| 79 | \( 1 - 1.31e19T + 8.96e37T^{2} \) |
| 83 | \( 1 + 1.15e19iT - 2.40e38T^{2} \) |
| 89 | \( 1 - 6.32e18iT - 9.72e38T^{2} \) |
| 97 | \( 1 + 6.37e18T + 5.43e39T^{2} \) |
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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
−11.21773400555107761393772296656, −9.905939717279879514050084111279, −9.007408635296236949374266367986, −8.122021393637029993385848128337, −6.25324311289078920883591288181, −5.26671804463660670820414754978, −3.82969565121854054711025776072, −3.03914595714195134963667992164, −1.31119556706001566639719443853, −0.67987750877347214129506989961,
0.63577403282183119974548151830, 2.49625334849143654502158031556, 3.46047407042212188654914882376, 4.82236859924331051743401726404, 6.47869148896908066979596755938, 6.73856231755160909632674183621, 8.047104515203580228370956892631, 9.597403229728244866163449325451, 10.21360812079084602596268853225, 11.68621007280651139702275438056
