| 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 + 8·23-s + 24-s + 25-s − 26-s + 27-s + 2·28-s − 6·29-s − 30-s + 4·31-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 + 1.66·23-s + 0.204·24-s + 1/5·25-s − 0.196·26-s + 0.192·27-s + 0.377·28-s − 1.11·29-s − 0.182·30-s + 0.718·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.09798297819972, −13.32545716681518, −12.99714540767384, −12.54965573123229, −11.82707683911093, −11.62619276592521, −11.00035468041373, −10.66473896641991, −9.977374897710050, −9.401971310331825, −8.940749036422795, −8.271397115334593, −7.963583927116264, −7.325945432154648, −6.973969597128665, −6.440327622095510, −5.547585869406869, −5.189902944269620, −4.618652894296924, −4.203628591525256, −3.416826846138258, −3.068329489673230, −2.444119701669700, −1.609562196323628, −1.190862978145358, 0,
1.190862978145358, 1.609562196323628, 2.444119701669700, 3.068329489673230, 3.416826846138258, 4.203628591525256, 4.618652894296924, 5.189902944269620, 5.547585869406869, 6.440327622095510, 6.973969597128665, 7.325945432154648, 7.963583927116264, 8.271397115334593, 8.940749036422795, 9.401971310331825, 9.977374897710050, 10.66473896641991, 11.00035468041373, 11.62619276592521, 11.82707683911093, 12.54965573123229, 12.99714540767384, 13.32545716681518, 14.09798297819972
