| L(s) = 1 | − 3-s + 9-s − 2·13-s + 6·17-s − 2·19-s − 27-s − 4·31-s − 6·37-s + 2·39-s + 2·41-s − 10·43-s − 2·47-s − 6·51-s + 6·53-s + 2·57-s + 12·59-s + 10·61-s − 10·67-s − 6·71-s − 10·73-s + 4·79-s + 81-s + 4·83-s − 6·89-s + 4·93-s − 2·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 0.554·13-s + 1.45·17-s − 0.458·19-s − 0.192·27-s − 0.718·31-s − 0.986·37-s + 0.320·39-s + 0.312·41-s − 1.52·43-s − 0.291·47-s − 0.840·51-s + 0.824·53-s + 0.264·57-s + 1.56·59-s + 1.28·61-s − 1.22·67-s − 0.712·71-s − 1.17·73-s + 0.450·79-s + 1/9·81-s + 0.439·83-s − 0.635·89-s + 0.414·93-s − 0.203·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 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
−14.54165800361942, −14.24911358093577, −13.46689864826124, −12.98778061556104, −12.56823902174966, −11.90382601730465, −11.72747825421294, −11.07135093245331, −10.38051493375541, −10.04546260951768, −9.703734163884049, −8.740474398217290, −8.539667475872337, −7.646184584166854, −7.283013166450229, −6.774627199305558, −6.020317468548420, −5.584097539755330, −5.042501872521200, −4.510630301166531, −3.662465609501234, −3.281927792752530, −2.336926925599582, −1.683687970322015, −0.8695334885819742, 0,
0.8695334885819742, 1.683687970322015, 2.336926925599582, 3.281927792752530, 3.662465609501234, 4.510630301166531, 5.042501872521200, 5.584097539755330, 6.020317468548420, 6.774627199305558, 7.283013166450229, 7.646184584166854, 8.539667475872337, 8.740474398217290, 9.703734163884049, 10.04546260951768, 10.38051493375541, 11.07135093245331, 11.72747825421294, 11.90382601730465, 12.56823902174966, 12.98778061556104, 13.46689864826124, 14.24911358093577, 14.54165800361942
