| L(s) = 1 | + 2.30·3-s + 1.30·5-s + 2.30·9-s + 4·11-s + 4.60·13-s + 3·15-s + 17-s − 8.60·19-s + 4·23-s − 3.30·25-s − 1.60·27-s + 9.21·29-s − 7.30·31-s + 9.21·33-s + 9.81·37-s + 10.6·39-s + 11.5·41-s − 4.30·43-s + 3.00·45-s + 2.60·47-s + 2.30·51-s + 0.697·53-s + 5.21·55-s − 19.8·57-s + 8·59-s − 15.5·61-s + 6·65-s + ⋯ |
| L(s) = 1 | + 1.32·3-s + 0.582·5-s + 0.767·9-s + 1.20·11-s + 1.27·13-s + 0.774·15-s + 0.242·17-s − 1.97·19-s + 0.834·23-s − 0.660·25-s − 0.308·27-s + 1.71·29-s − 1.31·31-s + 1.60·33-s + 1.61·37-s + 1.69·39-s + 1.79·41-s − 0.656·43-s + 0.447·45-s + 0.380·47-s + 0.322·51-s + 0.0957·53-s + 0.702·55-s − 2.62·57-s + 1.04·59-s − 1.98·61-s + 0.744·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.823631092\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.823631092\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 - 2.30T + 3T^{2} \) |
| 5 | \( 1 - 1.30T + 5T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 - 4.60T + 13T^{2} \) |
| 19 | \( 1 + 8.60T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 - 9.21T + 29T^{2} \) |
| 31 | \( 1 + 7.30T + 31T^{2} \) |
| 37 | \( 1 - 9.81T + 37T^{2} \) |
| 41 | \( 1 - 11.5T + 41T^{2} \) |
| 43 | \( 1 + 4.30T + 43T^{2} \) |
| 47 | \( 1 - 2.60T + 47T^{2} \) |
| 53 | \( 1 - 0.697T + 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 + 15.5T + 61T^{2} \) |
| 67 | \( 1 - 2.69T + 67T^{2} \) |
| 71 | \( 1 + 3.39T + 71T^{2} \) |
| 73 | \( 1 + 7.51T + 73T^{2} \) |
| 79 | \( 1 + 2.60T + 79T^{2} \) |
| 83 | \( 1 + 3.21T + 83T^{2} \) |
| 89 | \( 1 + 7.81T + 89T^{2} \) |
| 97 | \( 1 - 13.3T + 97T^{2} \) |
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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
−8.718648842119222093580900860835, −8.147484629416981854397868463540, −7.21640803018341674063951741026, −6.28534450421200033161846546478, −5.91757040486787312808952087741, −4.41241946058535657515881015690, −3.91523616946610771549225298010, −2.97928863720325613192907106650, −2.11828384774845839492676935403, −1.21959480004910906126175542136,
1.21959480004910906126175542136, 2.11828384774845839492676935403, 2.97928863720325613192907106650, 3.91523616946610771549225298010, 4.41241946058535657515881015690, 5.91757040486787312808952087741, 6.28534450421200033161846546478, 7.21640803018341674063951741026, 8.147484629416981854397868463540, 8.718648842119222093580900860835
