| L(s) = 1 | − 8i·2-s − 64·4-s + 197. i·5-s + 828. i·7-s + 512i·8-s + 1.57e3·10-s − 2.52e3i·11-s + (−239. − 7.91e3i)13-s + 6.62e3·14-s + 4.09e3·16-s − 1.67e4·17-s − 4.51e4i·19-s − 1.26e4i·20-s − 2.02e4·22-s − 5.36e4·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s + 0.706i·5-s + 0.912i·7-s + 0.353i·8-s + 0.499·10-s − 0.572i·11-s + (−0.0302 − 0.999i)13-s + 0.645·14-s + 0.250·16-s − 0.826·17-s − 1.51i·19-s − 0.353i·20-s − 0.404·22-s − 0.920·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0302i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0302i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.697419626\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.697419626\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 8iT \) |
| 3 | \( 1 \) |
| 13 | \( 1 + (239. + 7.91e3i)T \) |
| good | 5 | \( 1 - 197. iT - 7.81e4T^{2} \) |
| 7 | \( 1 - 828. iT - 8.23e5T^{2} \) |
| 11 | \( 1 + 2.52e3iT - 1.94e7T^{2} \) |
| 17 | \( 1 + 1.67e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 4.51e4iT - 8.93e8T^{2} \) |
| 23 | \( 1 + 5.36e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.57e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.28e5iT - 2.75e10T^{2} \) |
| 37 | \( 1 - 9.56e4iT - 9.49e10T^{2} \) |
| 41 | \( 1 - 8.44e5iT - 1.94e11T^{2} \) |
| 43 | \( 1 - 9.17e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 5.97e5iT - 5.06e11T^{2} \) |
| 53 | \( 1 + 5.76e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.63e6iT - 2.48e12T^{2} \) |
| 61 | \( 1 - 1.55e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 8.24e5iT - 6.06e12T^{2} \) |
| 71 | \( 1 - 5.63e6iT - 9.09e12T^{2} \) |
| 73 | \( 1 + 2.72e6iT - 1.10e13T^{2} \) |
| 79 | \( 1 + 3.75e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 1.52e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 - 1.74e6iT - 4.42e13T^{2} \) |
| 97 | \( 1 - 1.06e7iT - 8.07e13T^{2} \) |
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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
−10.96785367649612617708674962736, −10.16734257403743022827289175157, −9.001612413426898609134828398437, −8.259857707877860120937311554939, −6.82983587027317691980176951809, −5.74999131417956477282142670908, −4.57926514130120446325305835511, −3.02481915747946150820317717286, −2.49456389513205205612119693509, −0.793751423629071173132837265747,
0.57037342486153059289766469848, 1.89632325012941739919426921886, 3.98549315011069810658788984988, 4.56029033273465853590982664543, 5.92115544115684914523009696089, 6.95667807929009909613781371402, 7.86220763891860807578057217838, 8.864642111487069548427301716278, 9.780721900121122604093587968789, 10.75123846585270601298386268427
