| L(s) = 1 | − 8i·2-s − 64·4-s + 477. i·5-s − 855. i·7-s + 512i·8-s + 3.81e3·10-s − 1.40e3i·11-s + (5.68e3 − 5.51e3i)13-s − 6.84e3·14-s + 4.09e3·16-s − 2.45e4·17-s + 1.68e4i·19-s − 3.05e4i·20-s − 1.12e4·22-s + 5.41e4·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s + 1.70i·5-s − 0.942i·7-s + 0.353i·8-s + 1.20·10-s − 0.319i·11-s + (0.717 − 0.696i)13-s − 0.666·14-s + 0.250·16-s − 1.21·17-s + 0.564i·19-s − 0.853i·20-s − 0.225·22-s + 0.928·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.696 + 0.717i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.696 + 0.717i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.767674824\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.767674824\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 + (-5.68e3 + 5.51e3i)T \) |
| good | 5 | \( 1 - 477. iT - 7.81e4T^{2} \) |
| 7 | \( 1 + 855. iT - 8.23e5T^{2} \) |
| 11 | \( 1 + 1.40e3iT - 1.94e7T^{2} \) |
| 17 | \( 1 + 2.45e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.68e4iT - 8.93e8T^{2} \) |
| 23 | \( 1 - 5.41e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.80e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 7.30e4iT - 2.75e10T^{2} \) |
| 37 | \( 1 + 3.10e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 + 4.05e5iT - 1.94e11T^{2} \) |
| 43 | \( 1 - 6.26e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.19e6iT - 5.06e11T^{2} \) |
| 53 | \( 1 - 1.10e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.21e6iT - 2.48e12T^{2} \) |
| 61 | \( 1 - 2.71e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.35e6iT - 6.06e12T^{2} \) |
| 71 | \( 1 + 2.11e6iT - 9.09e12T^{2} \) |
| 73 | \( 1 + 2.97e5iT - 1.10e13T^{2} \) |
| 79 | \( 1 - 6.13e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 5.86e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 - 5.02e6iT - 4.42e13T^{2} \) |
| 97 | \( 1 + 4.56e6iT - 8.07e13T^{2} \) |
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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
−10.99181137325923248789430095684, −10.24520696914756588198470678213, −9.046753104656428790516572214536, −7.69327253035340982250962977952, −6.87072375837277624998316113103, −5.75460051170511312721233437391, −4.01673871752750253462398194618, −3.29414867743035764599868339449, −2.15992412938200714332180062184, −0.60747449243357315230173779986,
0.77485044444179829507354481768, 2.07963697575632900463823860263, 4.09198338136029062758816910553, 4.96709152117567926544529511965, 5.81041946890684573109552454134, 7.00377693875865975497953009652, 8.403985371547359171360480822739, 8.912590213676093139364615401960, 9.506137178082440901471142463519, 11.24861990626084452837391047422
