| L(s) = 1 | + (−42.2 + 20.0i)3-s − 125i·5-s + 1.61e3i·7-s + (1.37e3 − 1.69e3i)9-s − 504.·11-s − 9.49e3·13-s + (2.51e3 + 5.27e3i)15-s − 4.16e3i·17-s + 4.38e4i·19-s + (−3.24e4 − 6.82e4i)21-s + 2.23e4·23-s − 1.56e4·25-s + (−2.41e4 + 9.93e4i)27-s + 2.44e5i·29-s − 5.62e4i·31-s + ⋯ |
| L(s) = 1 | + (−0.902 + 0.429i)3-s − 0.447i·5-s + 1.77i·7-s + (0.630 − 0.775i)9-s − 0.114·11-s − 1.19·13-s + (0.192 + 0.403i)15-s − 0.205i·17-s + 1.46i·19-s + (−0.764 − 1.60i)21-s + 0.383·23-s − 0.199·25-s + (−0.236 + 0.971i)27-s + 1.86i·29-s − 0.339i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.567 + 0.823i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.567 + 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.2577222288\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2577222288\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (42.2 - 20.0i)T \) |
| 5 | \( 1 + 125iT \) |
| good | 7 | \( 1 - 1.61e3iT - 8.23e5T^{2} \) |
| 11 | \( 1 + 504.T + 1.94e7T^{2} \) |
| 13 | \( 1 + 9.49e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 4.16e3iT - 4.10e8T^{2} \) |
| 19 | \( 1 - 4.38e4iT - 8.93e8T^{2} \) |
| 23 | \( 1 - 2.23e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 2.44e5iT - 1.72e10T^{2} \) |
| 31 | \( 1 + 5.62e4iT - 2.75e10T^{2} \) |
| 37 | \( 1 - 2.55e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 7.04e4iT - 1.94e11T^{2} \) |
| 43 | \( 1 - 2.40e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 - 5.63e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.03e6iT - 1.17e12T^{2} \) |
| 59 | \( 1 + 2.42e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.04e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 9.30e4iT - 6.06e12T^{2} \) |
| 71 | \( 1 + 5.15e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.53e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 6.21e6iT - 1.92e13T^{2} \) |
| 83 | \( 1 - 9.59e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 9.31e6iT - 4.42e13T^{2} \) |
| 97 | \( 1 - 9.65e5T + 8.07e13T^{2} \) |
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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
−11.74810460562207743279780022929, −10.54233914790877299093251108193, −9.525596316531785793871945175598, −8.865931731999034249789281303483, −7.54720491008424058382829000186, −6.12520230303132060541843856934, −5.42460548646659116617064345039, −4.60413993264008508531882922710, −2.97512214108888258908136564579, −1.57791578142911233161426949434,
0.085652503590857423268690371271, 0.900178725333077761205268349766, 2.45733052016288518910220314788, 4.11526824271605844562389518513, 4.98107155517121817558017798920, 6.42777657222255392719952393062, 7.19597066098583999553780368379, 7.77812078802319043572549845931, 9.627492565564584494931957444507, 10.45728485335317053519255219646
