| L(s) = 1 | − 3-s − 5-s − 2·9-s + 3·11-s + 2·13-s + 15-s + 17-s − 3·19-s − 5·23-s − 4·25-s + 5·27-s − 2·29-s + 5·31-s − 3·33-s + 7·37-s − 2·39-s + 41-s + 4·43-s + 2·45-s − 3·47-s − 51-s − 3·53-s − 3·55-s + 3·57-s + 5·59-s + 3·61-s − 2·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 2/3·9-s + 0.904·11-s + 0.554·13-s + 0.258·15-s + 0.242·17-s − 0.688·19-s − 1.04·23-s − 4/5·25-s + 0.962·27-s − 0.371·29-s + 0.898·31-s − 0.522·33-s + 1.15·37-s − 0.320·39-s + 0.156·41-s + 0.609·43-s + 0.298·45-s − 0.437·47-s − 0.140·51-s − 0.412·53-s − 0.404·55-s + 0.397·57-s + 0.650·59-s + 0.384·61-s − 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 - 3 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 5 T + p T^{2} \) |
| 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
−7.55590452869201958633315923661, −6.57713248028730118761761587051, −6.10498045905791879611741409960, −5.61122146135671921471852942608, −4.51764633561710404537496213998, −4.03536608735973205947435767978, −3.20358972712684704955169742879, −2.19649705705242950706124745292, −1.09617751502925079956797898004, 0,
1.09617751502925079956797898004, 2.19649705705242950706124745292, 3.20358972712684704955169742879, 4.03536608735973205947435767978, 4.51764633561710404537496213998, 5.61122146135671921471852942608, 6.10498045905791879611741409960, 6.57713248028730118761761587051, 7.55590452869201958633315923661
