| L(s) = 1 | − 3-s − 3·5-s + 2·7-s − 2·9-s + 11-s + 3·15-s + 2·19-s − 2·21-s + 3·23-s + 4·25-s + 5·27-s − 6·29-s − 31-s − 33-s − 6·35-s + 7·37-s − 6·41-s − 8·43-s + 6·45-s + 12·47-s − 3·49-s − 6·53-s − 3·55-s − 2·57-s + 9·59-s + 2·61-s − 4·63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34·5-s + 0.755·7-s − 2/3·9-s + 0.301·11-s + 0.774·15-s + 0.458·19-s − 0.436·21-s + 0.625·23-s + 4/5·25-s + 0.962·27-s − 1.11·29-s − 0.179·31-s − 0.174·33-s − 1.01·35-s + 1.15·37-s − 0.937·41-s − 1.21·43-s + 0.894·45-s + 1.75·47-s − 3/7·49-s − 0.824·53-s − 0.404·55-s − 0.264·57-s + 1.17·59-s + 0.256·61-s − 0.503·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−15.33812800072061, −14.82364780763583, −14.62189901049541, −13.88592675824639, −13.21652554465863, −12.66696698672303, −11.89406534509193, −11.62488956520167, −11.39307051816054, −10.77151652772865, −10.24197710518164, −9.304092896083915, −8.859753491551916, −8.191658878147546, −7.799589625793655, −7.210700185955551, −6.630277521275154, −5.830175092778275, −5.280924077954580, −4.705798362705862, −4.081373630891592, −3.432300425905840, −2.778704181133429, −1.731004201626380, −0.8364033617383222, 0,
0.8364033617383222, 1.731004201626380, 2.778704181133429, 3.432300425905840, 4.081373630891592, 4.705798362705862, 5.280924077954580, 5.830175092778275, 6.630277521275154, 7.210700185955551, 7.799589625793655, 8.191658878147546, 8.859753491551916, 9.304092896083915, 10.24197710518164, 10.77151652772865, 11.39307051816054, 11.62488956520167, 11.89406534509193, 12.66696698672303, 13.21652554465863, 13.88592675824639, 14.62189901049541, 14.82364780763583, 15.33812800072061
