| L(s) = 1 | + 27i·3-s + 446. i·5-s − 1.63e3·7-s − 729·9-s + 7.53e3i·11-s + 5.16e3i·13-s − 1.20e4·15-s + 2.93e3·17-s + 4.69e4i·19-s − 4.40e4i·21-s − 5.06e4·23-s − 1.20e5·25-s − 1.96e4i·27-s + 1.64e4i·29-s − 4.18e4·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 1.59i·5-s − 1.79·7-s − 0.333·9-s + 1.70i·11-s + 0.652i·13-s − 0.921·15-s + 0.144·17-s + 1.57i·19-s − 1.03i·21-s − 0.867·23-s − 1.54·25-s − 0.192i·27-s + 0.125i·29-s − 0.252·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.062219738\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.062219738\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 27iT \) |
| good | 5 | \( 1 - 446. iT - 7.81e4T^{2} \) |
| 7 | \( 1 + 1.63e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 7.53e3iT - 1.94e7T^{2} \) |
| 13 | \( 1 - 5.16e3iT - 6.27e7T^{2} \) |
| 17 | \( 1 - 2.93e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 4.69e4iT - 8.93e8T^{2} \) |
| 23 | \( 1 + 5.06e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.64e4iT - 1.72e10T^{2} \) |
| 31 | \( 1 + 4.18e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.42e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 - 6.42e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 5.72e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 - 8.04e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 5.29e5iT - 1.17e12T^{2} \) |
| 59 | \( 1 + 1.71e6iT - 2.48e12T^{2} \) |
| 61 | \( 1 + 2.91e6iT - 3.14e12T^{2} \) |
| 67 | \( 1 - 2.41e6iT - 6.06e12T^{2} \) |
| 71 | \( 1 + 2.16e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 2.11e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 2.20e5T + 1.92e13T^{2} \) |
| 83 | \( 1 - 3.13e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 - 3.97e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.67e7T + 8.07e13T^{2} \) |
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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
−11.96213230209037960897471179755, −10.64442805163340816061831578859, −9.890161200935962745596356084610, −9.549695384059310846522159534967, −7.64563177204369761044707159119, −6.70608903986203801904792521216, −6.02698777531524786653724008102, −4.15288389676032443684115668489, −3.27955693814581056919850915916, −2.19396382274528500510837380857,
0.41381793156184767354006607276, 0.68578258863824911962671792053, 2.67604775232997153224642225490, 3.86315833071470789884345135574, 5.51486644615718817831920925696, 6.14569694672377699235101011914, 7.52234844992556282154179885777, 8.816214279499726625981470673754, 9.141943847776307190492756134966, 10.52293562967638533031634960419
