| L(s) = 1 | − 3-s + 5·5-s + 7·7-s − 26·9-s − 39·11-s − 17·13-s − 5·15-s − 15·17-s + 74·19-s − 7·21-s − 14·23-s + 25·25-s + 53·27-s − 237·29-s − 180·31-s + 39·33-s + 35·35-s − 318·37-s + 17·39-s − 348·41-s − 22·43-s − 130·45-s − 193·47-s + 49·49-s + 15·51-s − 208·53-s − 195·55-s + ⋯ |
| L(s) = 1 | − 0.192·3-s + 0.447·5-s + 0.377·7-s − 0.962·9-s − 1.06·11-s − 0.362·13-s − 0.0860·15-s − 0.214·17-s + 0.893·19-s − 0.0727·21-s − 0.126·23-s + 1/5·25-s + 0.377·27-s − 1.51·29-s − 1.04·31-s + 0.205·33-s + 0.169·35-s − 1.41·37-s + 0.0697·39-s − 1.32·41-s − 0.0780·43-s − 0.430·45-s − 0.598·47-s + 1/7·49-s + 0.0411·51-s − 0.539·53-s − 0.478·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - p T \) |
| 7 | \( 1 - p T \) |
| good | 3 | \( 1 + T + p^{3} T^{2} \) |
| 11 | \( 1 + 39 T + p^{3} T^{2} \) |
| 13 | \( 1 + 17 T + p^{3} T^{2} \) |
| 17 | \( 1 + 15 T + p^{3} T^{2} \) |
| 19 | \( 1 - 74 T + p^{3} T^{2} \) |
| 23 | \( 1 + 14 T + p^{3} T^{2} \) |
| 29 | \( 1 + 237 T + p^{3} T^{2} \) |
| 31 | \( 1 + 180 T + p^{3} T^{2} \) |
| 37 | \( 1 + 318 T + p^{3} T^{2} \) |
| 41 | \( 1 + 348 T + p^{3} T^{2} \) |
| 43 | \( 1 + 22 T + p^{3} T^{2} \) |
| 47 | \( 1 + 193 T + p^{3} T^{2} \) |
| 53 | \( 1 + 208 T + p^{3} T^{2} \) |
| 59 | \( 1 - 452 T + p^{3} T^{2} \) |
| 61 | \( 1 - 340 T + p^{3} T^{2} \) |
| 67 | \( 1 + 408 T + p^{3} T^{2} \) |
| 71 | \( 1 - 528 T + p^{3} T^{2} \) |
| 73 | \( 1 + 554 T + p^{3} T^{2} \) |
| 79 | \( 1 - 539 T + p^{3} T^{2} \) |
| 83 | \( 1 - 164 T + p^{3} T^{2} \) |
| 89 | \( 1 + 576 T + p^{3} T^{2} \) |
| 97 | \( 1 + 827 T + p^{3} T^{2} \) |
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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
−10.99733655032503238950331652097, −10.07836868033850112906066832593, −9.035044591075715380255396324543, −8.055701224541250542231443555946, −7.03778387840515664565140055806, −5.62126631695288955813181923240, −5.10728971523837573944579025726, −3.32087211172066782001659805168, −2.00122517121427611305092174829, 0,
2.00122517121427611305092174829, 3.32087211172066782001659805168, 5.10728971523837573944579025726, 5.62126631695288955813181923240, 7.03778387840515664565140055806, 8.055701224541250542231443555946, 9.035044591075715380255396324543, 10.07836868033850112906066832593, 10.99733655032503238950331652097
