| L(s) = 1 | + 2·3-s − 5-s + 9-s − 4·11-s − 13-s − 2·15-s + 2·17-s + 19-s − 7·23-s − 4·25-s − 4·27-s − 5·29-s + 9·31-s − 8·33-s − 2·37-s − 2·39-s − 2·41-s + 43-s − 45-s − 9·47-s + 4·51-s + 3·53-s + 4·55-s + 2·57-s − 14·61-s + 65-s + 10·67-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.447·5-s + 1/3·9-s − 1.20·11-s − 0.277·13-s − 0.516·15-s + 0.485·17-s + 0.229·19-s − 1.45·23-s − 4/5·25-s − 0.769·27-s − 0.928·29-s + 1.61·31-s − 1.39·33-s − 0.328·37-s − 0.320·39-s − 0.312·41-s + 0.152·43-s − 0.149·45-s − 1.31·47-s + 0.560·51-s + 0.412·53-s + 0.539·55-s + 0.264·57-s − 1.79·61-s + 0.124·65-s + 1.22·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + 5 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−8.237951521260646389256048043119, −7.952979788134185991537908763708, −7.38722006146820504716353926084, −6.18053403972595335678929800447, −5.35458385451819767395782385034, −4.35286193268314891819256010308, −3.47864491024103512148202589639, −2.74596489705447864537326689779, −1.85652707770394323743804698835, 0,
1.85652707770394323743804698835, 2.74596489705447864537326689779, 3.47864491024103512148202589639, 4.35286193268314891819256010308, 5.35458385451819767395782385034, 6.18053403972595335678929800447, 7.38722006146820504716353926084, 7.952979788134185991537908763708, 8.237951521260646389256048043119
