| L(s) = 1 | + (−77.3 − 24.0i)3-s + (−566. + 263. i)5-s + 3.44e3i·7-s + (5.40e3 + 3.72e3i)9-s − 1.54e4i·11-s − 9.93e3i·13-s + (5.01e4 − 6.72e3i)15-s − 4.29e4·17-s − 4.49e4·19-s + (8.29e4 − 2.66e5i)21-s − 2.96e5·23-s + (2.51e5 − 2.98e5i)25-s + (−3.28e5 − 4.17e5i)27-s − 6.28e5i·29-s − 1.30e6·31-s + ⋯ |
| L(s) = 1 | + (−0.954 − 0.297i)3-s + (−0.906 + 0.421i)5-s + 1.43i·7-s + (0.823 + 0.567i)9-s − 1.05i·11-s − 0.347i·13-s + (0.991 − 0.132i)15-s − 0.514·17-s − 0.344·19-s + (0.426 − 1.37i)21-s − 1.05·23-s + (0.645 − 0.764i)25-s + (−0.617 − 0.786i)27-s − 0.888i·29-s − 1.40·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.132i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.991 - 0.132i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.6123027934\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6123027934\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (77.3 + 24.0i)T \) |
| 5 | \( 1 + (566. - 263. i)T \) |
| good | 7 | \( 1 - 3.44e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 + 1.54e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 + 9.93e3iT - 8.15e8T^{2} \) |
| 17 | \( 1 + 4.29e4T + 6.97e9T^{2} \) |
| 19 | \( 1 + 4.49e4T + 1.69e10T^{2} \) |
| 23 | \( 1 + 2.96e5T + 7.83e10T^{2} \) |
| 29 | \( 1 + 6.28e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 + 1.30e6T + 8.52e11T^{2} \) |
| 37 | \( 1 - 1.56e6iT - 3.51e12T^{2} \) |
| 41 | \( 1 - 9.80e5iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 5.90e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 + 6.68e6T + 2.38e13T^{2} \) |
| 53 | \( 1 - 8.16e6T + 6.22e13T^{2} \) |
| 59 | \( 1 + 2.30e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 6.16e6T + 1.91e14T^{2} \) |
| 67 | \( 1 + 3.43e7iT - 4.06e14T^{2} \) |
| 71 | \( 1 + 4.22e6iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 1.17e7iT - 8.06e14T^{2} \) |
| 79 | \( 1 + 2.04e7T + 1.51e15T^{2} \) |
| 83 | \( 1 + 7.73e7T + 2.25e15T^{2} \) |
| 89 | \( 1 + 6.91e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + 7.42e7iT - 7.83e15T^{2} \) |
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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
−11.17521304715122885069586953157, −9.892794478305710720079703637555, −8.544449584784710984800911676691, −7.82126909189140545369949945795, −6.45908689384761214900982383602, −5.83813724278396883453555631772, −4.67571722966250890381327591310, −3.29811662783666855809127323036, −2.01419744349946887870349421311, −0.37469915448015138101239472945,
0.41805254469741794645986179418, 1.63089239085741891275969837983, 3.94358668711439262356785028201, 4.22802298546698343745182637106, 5.40826974655209394316583378430, 7.02589574893644398371399849352, 7.27742403177087229021890196323, 8.776240259774379565229433365515, 9.995813175321197178124933942832, 10.70360809989543659578240319953
