| L(s) = 1 | − 27i·3-s + (−50 − 275i)5-s − 1.12e3i·7-s − 729·9-s + 5.51e3·11-s + 1.27e4i·13-s + (−7.42e3 + 1.35e3i)15-s + 3.22e4i·17-s − 4.44e3·19-s − 3.04e4·21-s + 9.54e4i·23-s + (−7.31e4 + 2.75e4i)25-s + 1.96e4i·27-s − 1.94e4·29-s + 2.40e5·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + (−0.178 − 0.983i)5-s − 1.24i·7-s − 0.333·9-s + 1.24·11-s + 1.61i·13-s + (−0.568 + 0.103i)15-s + 1.58i·17-s − 0.148·19-s − 0.716·21-s + 1.63i·23-s + (−0.935 + 0.351i)25-s + 0.192i·27-s − 0.148·29-s + 1.44·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.178i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.983 - 0.178i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.790612024\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.790612024\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 27iT \) |
| 5 | \( 1 + (50 + 275i)T \) |
| good | 7 | \( 1 + 1.12e3iT - 8.23e5T^{2} \) |
| 11 | \( 1 - 5.51e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.27e4iT - 6.27e7T^{2} \) |
| 17 | \( 1 - 3.22e4iT - 4.10e8T^{2} \) |
| 19 | \( 1 + 4.44e3T + 8.93e8T^{2} \) |
| 23 | \( 1 - 9.54e4iT - 3.40e9T^{2} \) |
| 29 | \( 1 + 1.94e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.40e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 7.78e4iT - 9.49e10T^{2} \) |
| 41 | \( 1 - 2.99e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 4.16e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 + 3.22e5iT - 5.06e11T^{2} \) |
| 53 | \( 1 - 8.80e5iT - 1.17e12T^{2} \) |
| 59 | \( 1 + 1.84e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 8.61e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 6.73e5iT - 6.06e12T^{2} \) |
| 71 | \( 1 - 3.42e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 4.67e6iT - 1.10e13T^{2} \) |
| 79 | \( 1 + 3.13e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 4.84e5iT - 2.71e13T^{2} \) |
| 89 | \( 1 + 6.25e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 8.65e6iT - 8.07e13T^{2} \) |
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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
−11.14931798973250606745281236462, −9.766915284900454074061281284781, −8.931319039748286747761811299794, −7.923303658622125574836983466099, −6.92947250545652704520116508396, −6.02989049755201405842633898233, −4.37758051169004290143331145397, −3.84438424475241208460636317415, −1.62500074384728403654676001951, −1.11370687485318844825772264989,
0.47572912326310605033162534962, 2.50985481554923884598977710924, 3.18236816214765662281668435874, 4.62085819448578978027097544718, 5.81282741096414993949558300298, 6.68910779745843748694578937287, 8.012834059151936220076938224074, 9.000419861640643039304646869310, 9.908196050424788283761642943052, 10.83057860108615198882948220536
