| L(s) = 1 | + 5-s + 4·11-s + 2·13-s + 6·17-s − 4·19-s + 23-s + 25-s − 2·29-s + 2·37-s − 10·41-s + 4·43-s − 7·49-s + 6·53-s + 4·55-s − 4·59-s + 10·61-s + 2·65-s + 12·67-s + 8·71-s + 10·73-s − 8·79-s − 4·83-s + 6·85-s − 18·89-s − 4·95-s + 2·97-s + 101-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.20·11-s + 0.554·13-s + 1.45·17-s − 0.917·19-s + 0.208·23-s + 1/5·25-s − 0.371·29-s + 0.328·37-s − 1.56·41-s + 0.609·43-s − 49-s + 0.824·53-s + 0.539·55-s − 0.520·59-s + 1.28·61-s + 0.248·65-s + 1.46·67-s + 0.949·71-s + 1.17·73-s − 0.900·79-s − 0.439·83-s + 0.650·85-s − 1.90·89-s − 0.410·95-s + 0.203·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 66240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.421618764\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.421618764\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 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
−14.07409662314174, −13.95195102064758, −13.06741364908826, −12.79031625843100, −12.19281851159892, −11.70230373798662, −11.16619227685546, −10.73149036047558, −9.922172993062462, −9.767126257906991, −9.144982716760192, −8.400698479332227, −8.299663908072936, −7.390219182491916, −6.803104363374411, −6.425726569788953, −5.779218127007744, −5.337675442833463, −4.625735881974783, −3.843555884365168, −3.566271101559334, −2.754983911049880, −1.924977185902809, −1.366143028772428, −0.6571973606871682,
0.6571973606871682, 1.366143028772428, 1.924977185902809, 2.754983911049880, 3.566271101559334, 3.843555884365168, 4.625735881974783, 5.337675442833463, 5.779218127007744, 6.425726569788953, 6.803104363374411, 7.390219182491916, 8.299663908072936, 8.400698479332227, 9.144982716760192, 9.767126257906991, 9.922172993062462, 10.73149036047558, 11.16619227685546, 11.70230373798662, 12.19281851159892, 12.79031625843100, 13.06741364908826, 13.95195102064758, 14.07409662314174
