| L(s) = 1 | − 164. i·5-s + 289. i·7-s − 48.4·11-s − 995. i·13-s − 3.58e3·17-s + 6.08e3·19-s + 1.10e4i·23-s − 1.13e4·25-s + 2.19e4i·29-s − 2.18e4i·31-s + 4.75e4·35-s − 5.01e4i·37-s − 1.20e5·41-s + 5.72e4·43-s + 1.92e5i·47-s + ⋯ |
| L(s) = 1 | − 1.31i·5-s + 0.843i·7-s − 0.0363·11-s − 0.453i·13-s − 0.729·17-s + 0.887·19-s + 0.908i·23-s − 0.729·25-s + 0.898i·29-s − 0.731i·31-s + 1.10·35-s − 0.989i·37-s − 1.75·41-s + 0.719·43-s + 1.84i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.855655072\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.855655072\) |
| \(L(4)\) |
|
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 | \( 1 + 164. iT - 1.56e4T^{2} \) |
| 7 | \( 1 - 289. iT - 1.17e5T^{2} \) |
| 11 | \( 1 + 48.4T + 1.77e6T^{2} \) |
| 13 | \( 1 + 995. iT - 4.82e6T^{2} \) |
| 17 | \( 1 + 3.58e3T + 2.41e7T^{2} \) |
| 19 | \( 1 - 6.08e3T + 4.70e7T^{2} \) |
| 23 | \( 1 - 1.10e4iT - 1.48e8T^{2} \) |
| 29 | \( 1 - 2.19e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 + 2.18e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 + 5.01e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + 1.20e5T + 4.75e9T^{2} \) |
| 43 | \( 1 - 5.72e4T + 6.32e9T^{2} \) |
| 47 | \( 1 - 1.92e5iT - 1.07e10T^{2} \) |
| 53 | \( 1 - 1.04e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 - 5.83e4T + 4.21e10T^{2} \) |
| 61 | \( 1 - 2.01e5iT - 5.15e10T^{2} \) |
| 67 | \( 1 + 1.66e5T + 9.04e10T^{2} \) |
| 71 | \( 1 + 2.62e4iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 1.47e4T + 1.51e11T^{2} \) |
| 79 | \( 1 - 1.36e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 - 1.32e5T + 3.26e11T^{2} \) |
| 89 | \( 1 - 7.69e5T + 4.96e11T^{2} \) |
| 97 | \( 1 - 1.24e6T + 8.32e11T^{2} \) |
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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
−9.450770302593338508284296036714, −9.035410378809637259624975231192, −8.194695625477279365696593155915, −7.26010548126138566485923772333, −5.86413450055887642051116356389, −5.28431465260196710214548212307, −4.35332954215343735111853074339, −3.05993453808646230730226780057, −1.80251183169701962247306310009, −0.77101185830595860660758967077,
0.49687125520337006353280267292, 1.95992033554542990065747017302, 3.05044474742362288557085625363, 3.94239541216529319038438844458, 5.05658249985047219707272879686, 6.50840660032515549744101762214, 6.85997914437997184439712091606, 7.79016002351568084669692842324, 8.852265830338457170989133532865, 10.06157545246493249952361872175
