L(s) = 1 | + 2·3-s + 5·5-s + 6·7-s − 23·9-s − 60·11-s − 50·13-s + 10·15-s − 30·17-s − 40·19-s + 12·21-s + 178·23-s + 25·25-s − 100·27-s − 166·29-s + 20·31-s − 120·33-s + 30·35-s − 10·37-s − 100·39-s − 250·41-s − 142·43-s − 115·45-s + 214·47-s − 307·49-s − 60·51-s − 490·53-s − 300·55-s + ⋯ |
L(s) = 1 | + 0.384·3-s + 0.447·5-s + 0.323·7-s − 0.851·9-s − 1.64·11-s − 1.06·13-s + 0.172·15-s − 0.428·17-s − 0.482·19-s + 0.124·21-s + 1.61·23-s + 1/5·25-s − 0.712·27-s − 1.06·29-s + 0.115·31-s − 0.633·33-s + 0.144·35-s − 0.0444·37-s − 0.410·39-s − 0.952·41-s − 0.503·43-s − 0.380·45-s + 0.664·47-s − 0.895·49-s − 0.164·51-s − 1.26·53-s − 0.735·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p T \) |
good | 3 | \( 1 - 2 T + p^{3} T^{2} \) |
| 7 | \( 1 - 6 T + p^{3} T^{2} \) |
| 11 | \( 1 + 60 T + p^{3} T^{2} \) |
| 13 | \( 1 + 50 T + p^{3} T^{2} \) |
| 17 | \( 1 + 30 T + p^{3} T^{2} \) |
| 19 | \( 1 + 40 T + p^{3} T^{2} \) |
| 23 | \( 1 - 178 T + p^{3} T^{2} \) |
| 29 | \( 1 + 166 T + p^{3} T^{2} \) |
| 31 | \( 1 - 20 T + p^{3} T^{2} \) |
| 37 | \( 1 + 10 T + p^{3} T^{2} \) |
| 41 | \( 1 + 250 T + p^{3} T^{2} \) |
| 43 | \( 1 + 142 T + p^{3} T^{2} \) |
| 47 | \( 1 - 214 T + p^{3} T^{2} \) |
| 53 | \( 1 + 490 T + p^{3} T^{2} \) |
| 59 | \( 1 - 800 T + p^{3} T^{2} \) |
| 61 | \( 1 + 250 T + p^{3} T^{2} \) |
| 67 | \( 1 - 774 T + p^{3} T^{2} \) |
| 71 | \( 1 - 100 T + p^{3} T^{2} \) |
| 73 | \( 1 + 230 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1320 T + p^{3} T^{2} \) |
| 83 | \( 1 + 982 T + p^{3} T^{2} \) |
| 89 | \( 1 - 874 T + p^{3} T^{2} \) |
| 97 | \( 1 + 310 T + p^{3} T^{2} \) |
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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
−10.74117484326152692780562512006, −9.793474478394529685653776540025, −8.794232762529535560492863794752, −7.962455877217667446575635752774, −6.97420450074189405308819045722, −5.54338512834396441265237687964, −4.86092766598419121706399449507, −3.04798071184534646865842241607, −2.16752739821206801963148771763, 0,
2.16752739821206801963148771763, 3.04798071184534646865842241607, 4.86092766598419121706399449507, 5.54338512834396441265237687964, 6.97420450074189405308819045722, 7.962455877217667446575635752774, 8.794232762529535560492863794752, 9.793474478394529685653776540025, 10.74117484326152692780562512006