L(s) = 1 | − 3-s − 4·7-s + 9-s − 11-s − 2·13-s + 2·17-s + 4·21-s − 8·23-s − 27-s + 6·29-s − 8·31-s + 33-s + 6·37-s + 2·39-s − 2·41-s − 8·47-s + 9·49-s − 2·51-s + 6·53-s + 4·59-s − 6·61-s − 4·63-s − 4·67-s + 8·69-s + 14·73-s + 4·77-s − 4·79-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 1.51·7-s + 1/3·9-s − 0.301·11-s − 0.554·13-s + 0.485·17-s + 0.872·21-s − 1.66·23-s − 0.192·27-s + 1.11·29-s − 1.43·31-s + 0.174·33-s + 0.986·37-s + 0.320·39-s − 0.312·41-s − 1.16·47-s + 9/7·49-s − 0.280·51-s + 0.824·53-s + 0.520·59-s − 0.768·61-s − 0.503·63-s − 0.488·67-s + 0.963·69-s + 1.63·73-s + 0.455·77-s − 0.450·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 52800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−14.73572378873991, −14.19609416530022, −13.53905331459395, −13.15457494416960, −12.60969112414061, −12.11994723550853, −11.91315722457256, −11.04078955811806, −10.54509515567021, −10.01452192893950, −9.634370668082810, −9.283585528624298, −8.321040311416481, −7.933156653708424, −7.161460094699055, −6.775512363133880, −6.123800990136777, −5.772655780519693, −5.130799897107314, −4.402195027140824, −3.770888063849795, −3.198404523096050, −2.512134435758316, −1.771334806841168, −0.6750611742928177, 0,
0.6750611742928177, 1.771334806841168, 2.512134435758316, 3.198404523096050, 3.770888063849795, 4.402195027140824, 5.130799897107314, 5.772655780519693, 6.123800990136777, 6.775512363133880, 7.161460094699055, 7.933156653708424, 8.321040311416481, 9.283585528624298, 9.634370668082810, 10.01452192893950, 10.54509515567021, 11.04078955811806, 11.91315722457256, 12.11994723550853, 12.60969112414061, 13.15457494416960, 13.53905331459395, 14.19609416530022, 14.73572378873991