| L(s) = 1 | + 2·2-s + 4·4-s − 5·5-s − 22·7-s + 8·8-s − 10·10-s − 12·11-s + 38·13-s − 44·14-s + 16·16-s − 105·17-s − 157·19-s − 20·20-s − 24·22-s − 117·23-s + 25·25-s + 76·26-s − 88·28-s + 66·29-s − 25·31-s + 32·32-s − 210·34-s + 110·35-s + 314·37-s − 314·38-s − 40·40-s − 504·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s − 1.18·7-s + 0.353·8-s − 0.316·10-s − 0.328·11-s + 0.810·13-s − 0.839·14-s + 1/4·16-s − 1.49·17-s − 1.89·19-s − 0.223·20-s − 0.232·22-s − 1.06·23-s + 1/5·25-s + 0.573·26-s − 0.593·28-s + 0.422·29-s − 0.144·31-s + 0.176·32-s − 1.05·34-s + 0.531·35-s + 1.39·37-s − 1.34·38-s − 0.158·40-s − 1.91·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + p T \) |
| good | 7 | \( 1 + 22 T + p^{3} T^{2} \) |
| 11 | \( 1 + 12 T + p^{3} T^{2} \) |
| 13 | \( 1 - 38 T + p^{3} T^{2} \) |
| 17 | \( 1 + 105 T + p^{3} T^{2} \) |
| 19 | \( 1 + 157 T + p^{3} T^{2} \) |
| 23 | \( 1 + 117 T + p^{3} T^{2} \) |
| 29 | \( 1 - 66 T + p^{3} T^{2} \) |
| 31 | \( 1 + 25 T + p^{3} T^{2} \) |
| 37 | \( 1 - 314 T + p^{3} T^{2} \) |
| 41 | \( 1 + 504 T + p^{3} T^{2} \) |
| 43 | \( 1 - 380 T + p^{3} T^{2} \) |
| 47 | \( 1 + 252 T + p^{3} T^{2} \) |
| 53 | \( 1 - 3 T + p^{3} T^{2} \) |
| 59 | \( 1 + 318 T + p^{3} T^{2} \) |
| 61 | \( 1 - 293 T + p^{3} T^{2} \) |
| 67 | \( 1 + 322 T + p^{3} T^{2} \) |
| 71 | \( 1 + 120 T + p^{3} T^{2} \) |
| 73 | \( 1 - 44 T + p^{3} T^{2} \) |
| 79 | \( 1 - 917 T + p^{3} T^{2} \) |
| 83 | \( 1 - 309 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1272 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1328 T + p^{3} T^{2} \) |
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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
−11.05387228865946611274980054811, −10.32219244915996381938923026987, −9.028959659710674821844292966387, −8.047083257782184036089740260452, −6.60757288375329147996343051841, −6.19013651968810269528703903804, −4.54301646984761116174591765160, −3.65085861501872437041286609533, −2.31495434604174197069770530172, 0,
2.31495434604174197069770530172, 3.65085861501872437041286609533, 4.54301646984761116174591765160, 6.19013651968810269528703903804, 6.60757288375329147996343051841, 8.047083257782184036089740260452, 9.028959659710674821844292966387, 10.32219244915996381938923026987, 11.05387228865946611274980054811
