| L(s) = 1 | + 4·5-s − 3·11-s + 2·13-s − 2·17-s − 6·23-s + 11·25-s + 4·29-s − 10·31-s + 2·37-s − 9·41-s − 4·43-s + 12·47-s − 7·49-s + 2·53-s − 12·55-s + 59-s − 8·61-s + 8·65-s + 9·67-s + 6·71-s − 9·73-s − 4·79-s + 5·83-s − 8·85-s + 18·89-s + 97-s + 101-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 0.904·11-s + 0.554·13-s − 0.485·17-s − 1.25·23-s + 11/5·25-s + 0.742·29-s − 1.79·31-s + 0.328·37-s − 1.40·41-s − 0.609·43-s + 1.75·47-s − 49-s + 0.274·53-s − 1.61·55-s + 0.130·59-s − 1.02·61-s + 0.992·65-s + 1.09·67-s + 0.712·71-s − 1.05·73-s − 0.450·79-s + 0.548·83-s − 0.867·85-s + 1.90·89-s + 0.101·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25992 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25992 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 5 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−15.64988620496941, −14.94469761468871, −14.44313613906969, −13.78211563496270, −13.54652718206773, −13.06877523260421, −12.52213876004975, −11.89580191408297, −11.06964446677907, −10.51414715554653, −10.28084624246504, −9.544705214097904, −9.162801985331548, −8.482203905737661, −7.950856078925214, −7.098232464443712, −6.505590397816848, −5.993001418382951, −5.422539291764153, −5.016353810596496, −4.106794806785660, −3.281779417605337, −2.457227928960713, −1.988982405638203, −1.281774213568825, 0,
1.281774213568825, 1.988982405638203, 2.457227928960713, 3.281779417605337, 4.106794806785660, 5.016353810596496, 5.422539291764153, 5.993001418382951, 6.505590397816848, 7.098232464443712, 7.950856078925214, 8.482203905737661, 9.162801985331548, 9.544705214097904, 10.28084624246504, 10.51414715554653, 11.06964446677907, 11.89580191408297, 12.52213876004975, 13.06877523260421, 13.54652718206773, 13.78211563496270, 14.44313613906969, 14.94469761468871, 15.64988620496941
