| L(s) = 1 | + 2·2-s − 3-s + 4·4-s + 5·5-s − 2·6-s − 32·7-s + 8·8-s − 26·9-s + 10·10-s − 30·11-s − 4·12-s + 19·13-s − 64·14-s − 5·15-s + 16·16-s − 60·17-s − 52·18-s − 58·19-s + 20·20-s + 32·21-s − 60·22-s + 23·23-s − 8·24-s + 25·25-s + 38·26-s + 53·27-s − 128·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.192·3-s + 1/2·4-s + 0.447·5-s − 0.136·6-s − 1.72·7-s + 0.353·8-s − 0.962·9-s + 0.316·10-s − 0.822·11-s − 0.0962·12-s + 0.405·13-s − 1.22·14-s − 0.0860·15-s + 1/4·16-s − 0.856·17-s − 0.680·18-s − 0.700·19-s + 0.223·20-s + 0.332·21-s − 0.581·22-s + 0.208·23-s − 0.0680·24-s + 1/5·25-s + 0.286·26-s + 0.377·27-s − 0.863·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p T \) |
| 5 | \( 1 - p T \) |
| 23 | \( 1 - p T \) |
| good | 3 | \( 1 + T + p^{3} T^{2} \) |
| 7 | \( 1 + 32 T + p^{3} T^{2} \) |
| 11 | \( 1 + 30 T + p^{3} T^{2} \) |
| 13 | \( 1 - 19 T + p^{3} T^{2} \) |
| 17 | \( 1 + 60 T + p^{3} T^{2} \) |
| 19 | \( 1 + 58 T + p^{3} T^{2} \) |
| 29 | \( 1 - 85 T + p^{3} T^{2} \) |
| 31 | \( 1 + 65 T + p^{3} T^{2} \) |
| 37 | \( 1 + 34 T + p^{3} T^{2} \) |
| 41 | \( 1 - 143 T + p^{3} T^{2} \) |
| 43 | \( 1 + 332 T + p^{3} T^{2} \) |
| 47 | \( 1 + 561 T + p^{3} T^{2} \) |
| 53 | \( 1 + 422 T + p^{3} T^{2} \) |
| 59 | \( 1 - 392 T + p^{3} T^{2} \) |
| 61 | \( 1 + 246 T + p^{3} T^{2} \) |
| 67 | \( 1 - 894 T + p^{3} T^{2} \) |
| 71 | \( 1 + 737 T + p^{3} T^{2} \) |
| 73 | \( 1 - 1041 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1114 T + p^{3} T^{2} \) |
| 83 | \( 1 + 936 T + p^{3} T^{2} \) |
| 89 | \( 1 - 824 T + p^{3} T^{2} \) |
| 97 | \( 1 + 868 T + p^{3} T^{2} \) |
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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
−11.29920488299086513763960659894, −10.45152713829056279763477616302, −9.448241007486856623109048966701, −8.330578974549188690707441972149, −6.69430909669226178069965297405, −6.18026590069427707113719977824, −5.07336161382976973296069205904, −3.47698811522538188751092354733, −2.49769826450685197914044055914, 0,
2.49769826450685197914044055914, 3.47698811522538188751092354733, 5.07336161382976973296069205904, 6.18026590069427707113719977824, 6.69430909669226178069965297405, 8.330578974549188690707441972149, 9.448241007486856623109048966701, 10.45152713829056279763477616302, 11.29920488299086513763960659894
