| L(s) = 1 | − 8·3-s − 10·5-s − 16·7-s + 37·9-s + 40·11-s − 50·13-s + 80·15-s − 30·17-s − 40·19-s + 128·21-s − 48·23-s − 25·25-s − 80·27-s − 34·29-s − 320·31-s − 320·33-s + 160·35-s + 310·37-s + 400·39-s + 410·41-s − 152·43-s − 370·45-s + 416·47-s − 87·49-s + 240·51-s − 410·53-s − 400·55-s + ⋯ |
| L(s) = 1 | − 1.53·3-s − 0.894·5-s − 0.863·7-s + 1.37·9-s + 1.09·11-s − 1.06·13-s + 1.37·15-s − 0.428·17-s − 0.482·19-s + 1.33·21-s − 0.435·23-s − 1/5·25-s − 0.570·27-s − 0.217·29-s − 1.85·31-s − 1.68·33-s + 0.772·35-s + 1.37·37-s + 1.64·39-s + 1.56·41-s − 0.539·43-s − 1.22·45-s + 1.29·47-s − 0.253·49-s + 0.658·51-s − 1.06·53-s − 0.980·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 3 | \( 1 + 8 T + p^{3} T^{2} \) |
| 5 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 7 | \( 1 + 16 T + p^{3} T^{2} \) |
| 11 | \( 1 - 40 T + p^{3} T^{2} \) |
| 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 | \( 1 + 48 T + p^{3} T^{2} \) |
| 29 | \( 1 + 34 T + p^{3} T^{2} \) |
| 31 | \( 1 + 320 T + p^{3} T^{2} \) |
| 37 | \( 1 - 310 T + p^{3} T^{2} \) |
| 41 | \( 1 - 10 p T + p^{3} T^{2} \) |
| 43 | \( 1 + 152 T + p^{3} T^{2} \) |
| 47 | \( 1 - 416 T + p^{3} T^{2} \) |
| 53 | \( 1 + 410 T + p^{3} T^{2} \) |
| 59 | \( 1 - 200 T + p^{3} T^{2} \) |
| 61 | \( 1 - 30 T + p^{3} T^{2} \) |
| 67 | \( 1 + 776 T + p^{3} T^{2} \) |
| 71 | \( 1 + 400 T + p^{3} T^{2} \) |
| 73 | \( 1 + 630 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1120 T + p^{3} T^{2} \) |
| 83 | \( 1 + 552 T + p^{3} T^{2} \) |
| 89 | \( 1 + 326 T + p^{3} T^{2} \) |
| 97 | \( 1 + 110 T + p^{3} T^{2} \) |
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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
−16.10996357117449595775841857321, −14.77439041982288005015547383517, −12.76214242090854890131702355961, −11.93383977402877257653985877977, −10.97279434100881282480603513006, −9.476457375854773044333878361728, −7.26521015165622585731263708085, −6.03573278605739392477896149504, −4.23248986236507833679428303220, 0,
4.23248986236507833679428303220, 6.03573278605739392477896149504, 7.26521015165622585731263708085, 9.476457375854773044333878361728, 10.97279434100881282480603513006, 11.93383977402877257653985877977, 12.76214242090854890131702355961, 14.77439041982288005015547383517, 16.10996357117449595775841857321
