| L(s) = 1 | − 3-s + 9-s − 6·13-s + 2·17-s − 8·19-s + 8·23-s − 27-s − 2·29-s + 4·31-s + 2·37-s + 6·39-s + 6·41-s + 4·43-s − 8·47-s − 2·51-s − 10·53-s + 8·57-s + 4·59-s + 2·61-s + 4·67-s − 8·69-s + 12·71-s − 2·73-s − 8·79-s + 81-s + 4·83-s + 2·87-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 1.66·13-s + 0.485·17-s − 1.83·19-s + 1.66·23-s − 0.192·27-s − 0.371·29-s + 0.718·31-s + 0.328·37-s + 0.960·39-s + 0.937·41-s + 0.609·43-s − 1.16·47-s − 0.280·51-s − 1.37·53-s + 1.05·57-s + 0.520·59-s + 0.256·61-s + 0.488·67-s − 0.963·69-s + 1.42·71-s − 0.234·73-s − 0.900·79-s + 1/9·81-s + 0.439·83-s + 0.214·87-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 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−14.61593535222233, −14.33733035521996, −13.34744145626515, −13.01637281425753, −12.43770024503799, −12.26508813512266, −11.48199529181625, −10.95226620602164, −10.68574674374604, −9.920781100534557, −9.546943640340210, −9.067472034772326, −8.202982953387435, −7.905252410776016, −7.086137409560652, −6.784333575762370, −6.202173972493720, −5.476238013195452, −4.971093496378076, −4.505401806493081, −3.922203121718221, −2.949427931679554, −2.493268692406329, −1.717249155018236, −0.7960587340447650, 0,
0.7960587340447650, 1.717249155018236, 2.493268692406329, 2.949427931679554, 3.922203121718221, 4.505401806493081, 4.971093496378076, 5.476238013195452, 6.202173972493720, 6.784333575762370, 7.086137409560652, 7.905252410776016, 8.202982953387435, 9.067472034772326, 9.546943640340210, 9.920781100534557, 10.68574674374604, 10.95226620602164, 11.48199529181625, 12.26508813512266, 12.43770024503799, 13.01637281425753, 13.34744145626515, 14.33733035521996, 14.61593535222233
