| L(s) = 1 | − 2·3-s + 9-s − 3·11-s − 2·13-s − 4·17-s + 3·23-s + 4·27-s − 29-s − 2·31-s + 6·33-s − 7·37-s + 4·39-s − 2·41-s − 43-s − 12·47-s + 8·51-s − 6·53-s − 6·59-s + 6·61-s − 7·67-s − 6·69-s + 3·71-s + 2·73-s + 5·79-s − 11·81-s + 6·83-s + 2·87-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/3·9-s − 0.904·11-s − 0.554·13-s − 0.970·17-s + 0.625·23-s + 0.769·27-s − 0.185·29-s − 0.359·31-s + 1.04·33-s − 1.15·37-s + 0.640·39-s − 0.312·41-s − 0.152·43-s − 1.75·47-s + 1.12·51-s − 0.824·53-s − 0.781·59-s + 0.768·61-s − 0.855·67-s − 0.722·69-s + 0.356·71-s + 0.234·73-s + 0.562·79-s − 1.22·81-s + 0.658·83-s + 0.214·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−14.30397515842344, −13.63801262275009, −13.11779052074309, −12.80744359110024, −12.16179507519339, −11.81639518568726, −11.08918333028904, −10.99989655324677, −10.32243841180369, −9.973560249690357, −9.188387539961871, −8.787018917799634, −8.094643344356946, −7.582180317075741, −6.941855753680229, −6.514101824127571, −6.001812570516390, −5.320604983650629, −4.848575747075730, −4.684668380457902, −3.591671638402224, −3.072555032158506, −2.266128162017696, −1.667208514464751, −0.5986394789149848, 0,
0.5986394789149848, 1.667208514464751, 2.266128162017696, 3.072555032158506, 3.591671638402224, 4.684668380457902, 4.848575747075730, 5.320604983650629, 6.001812570516390, 6.514101824127571, 6.941855753680229, 7.582180317075741, 8.094643344356946, 8.787018917799634, 9.188387539961871, 9.973560249690357, 10.32243841180369, 10.99989655324677, 11.08918333028904, 11.81639518568726, 12.16179507519339, 12.80744359110024, 13.11779052074309, 13.63801262275009, 14.30397515842344
