| L(s) = 1 | + 24.7i·5-s − 50.9i·7-s − 139.·11-s − 98.9i·13-s + 16·17-s − 559.·19-s + 814. i·23-s + 13.0·25-s + 420. i·29-s + 661. i·31-s + 1.25e3·35-s − 2.47e3i·37-s + 464·41-s + 559.·43-s + 2.44e3i·47-s + ⋯ |
| L(s) = 1 | + 0.989i·5-s − 1.03i·7-s − 1.15·11-s − 0.585i·13-s + 0.0553·17-s − 1.55·19-s + 1.53i·23-s + 0.0208·25-s + 0.500i·29-s + 0.688i·31-s + 1.02·35-s − 1.80i·37-s + 0.276·41-s + 0.302·43-s + 1.10i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.479061521\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.479061521\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 24.7iT - 625T^{2} \) |
| 7 | \( 1 + 50.9iT - 2.40e3T^{2} \) |
| 11 | \( 1 + 139.T + 1.46e4T^{2} \) |
| 13 | \( 1 + 98.9iT - 2.85e4T^{2} \) |
| 17 | \( 1 - 16T + 8.35e4T^{2} \) |
| 19 | \( 1 + 559.T + 1.30e5T^{2} \) |
| 23 | \( 1 - 814. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 420. iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 661. iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 2.47e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 - 464T + 2.82e6T^{2} \) |
| 43 | \( 1 - 559.T + 3.41e6T^{2} \) |
| 47 | \( 1 - 2.44e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 3.19e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 3.63e3T + 1.21e7T^{2} \) |
| 61 | \( 1 + 4.25e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 - 5.59e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + 3.25e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 898T + 2.83e7T^{2} \) |
| 79 | \( 1 - 8.60e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 - 419.T + 4.74e7T^{2} \) |
| 89 | \( 1 - 7.07e3T + 6.27e7T^{2} \) |
| 97 | \( 1 + 2.36e3T + 8.85e7T^{2} \) |
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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
−9.378813196169852567450772875233, −8.206432375521951596733818878481, −7.51457813791916291851354557014, −6.91402757056995766020056881464, −5.93837423830918592002370043248, −4.97704966326612309142049178949, −3.84595758084896894922877514792, −3.05877987032950407806167101332, −1.99377752078594314505770008129, −0.51436029191417336001487739364,
0.57051986389146157631598174003, 2.01955753413185021238281899945, 2.72408432840358126403712821439, 4.31194476928256872328301443237, 4.88113683510336661385880493796, 5.83431673925106516963460771485, 6.57794150112910137953158300536, 7.86828089091010211291880189289, 8.589565744571473319503033373149, 8.954570255470139167723518337404
