| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 2·7-s − 8-s + 9-s + 10-s + 12-s − 13-s + 2·14-s − 15-s + 16-s − 18-s − 2·19-s − 20-s − 2·21-s − 6·23-s − 24-s + 25-s + 26-s + 27-s − 2·28-s + 30-s − 4·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.288·12-s − 0.277·13-s + 0.534·14-s − 0.258·15-s + 1/4·16-s − 0.235·18-s − 0.458·19-s − 0.223·20-s − 0.436·21-s − 1.25·23-s − 0.204·24-s + 1/5·25-s + 0.196·26-s + 0.192·27-s − 0.377·28-s + 0.182·30-s − 0.718·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−14.72004698344252, −14.52753831862184, −13.94018495149329, −13.13399792177558, −12.80201245037024, −12.32687440679529, −11.68705361871994, −11.19982465870188, −10.53584046051900, −10.09622425021871, −9.519686984076151, −9.181752885249687, −8.502683179009199, −8.047937167740602, −7.504043810984193, −7.065552907395597, −6.310869591732883, −5.968169369795323, −5.061008360270063, −4.271860952752683, −3.730352915701467, −3.138359904942572, −2.395605675438167, −1.874603499123495, −0.8151712429319077, 0,
0.8151712429319077, 1.874603499123495, 2.395605675438167, 3.138359904942572, 3.730352915701467, 4.271860952752683, 5.061008360270063, 5.968169369795323, 6.310869591732883, 7.065552907395597, 7.504043810984193, 8.047937167740602, 8.502683179009199, 9.181752885249687, 9.519686984076151, 10.09622425021871, 10.53584046051900, 11.19982465870188, 11.68705361871994, 12.32687440679529, 12.80201245037024, 13.13399792177558, 13.94018495149329, 14.52753831862184, 14.72004698344252
