| L(s) = 1 | − 2·3-s + 5-s + 9-s − 4·11-s − 2·13-s − 2·15-s − 2·17-s + 25-s + 4·27-s + 4·29-s + 8·31-s + 8·33-s − 2·37-s + 4·39-s + 12·41-s + 4·43-s + 45-s − 7·49-s + 4·51-s − 6·53-s − 4·55-s + 4·59-s − 6·61-s − 2·65-s + 10·67-s − 2·73-s − 2·75-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.447·5-s + 1/3·9-s − 1.20·11-s − 0.554·13-s − 0.516·15-s − 0.485·17-s + 1/5·25-s + 0.769·27-s + 0.742·29-s + 1.43·31-s + 1.39·33-s − 0.328·37-s + 0.640·39-s + 1.87·41-s + 0.609·43-s + 0.149·45-s − 49-s + 0.560·51-s − 0.824·53-s − 0.539·55-s + 0.520·59-s − 0.768·61-s − 0.248·65-s + 1.22·67-s − 0.234·73-s − 0.230·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−7.55777532172612001285260518339, −6.59244433373055151745378435722, −6.22661192181383419430640314282, −5.38552063681883890906316922538, −4.95167528872057331442748088585, −4.26534666359207137996064685283, −2.90328819250179221469195423070, −2.37103653324555701402397922357, −1.03248145630946355296636213050, 0,
1.03248145630946355296636213050, 2.37103653324555701402397922357, 2.90328819250179221469195423070, 4.26534666359207137996064685283, 4.95167528872057331442748088585, 5.38552063681883890906316922538, 6.22661192181383419430640314282, 6.59244433373055151745378435722, 7.55777532172612001285260518339
