| L(s) = 1 | − 2·3-s − 2·7-s + 9-s + 4·11-s + 13-s − 2·17-s + 4·21-s + 6·23-s + 4·27-s − 10·29-s − 8·33-s − 10·37-s − 2·39-s − 2·41-s − 2·43-s + 6·47-s − 3·49-s + 4·51-s − 2·53-s − 8·59-s + 2·61-s − 2·63-s + 6·67-s − 12·69-s − 8·71-s − 10·73-s − 8·77-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.755·7-s + 1/3·9-s + 1.20·11-s + 0.277·13-s − 0.485·17-s + 0.872·21-s + 1.25·23-s + 0.769·27-s − 1.85·29-s − 1.39·33-s − 1.64·37-s − 0.320·39-s − 0.312·41-s − 0.304·43-s + 0.875·47-s − 3/7·49-s + 0.560·51-s − 0.274·53-s − 1.04·59-s + 0.256·61-s − 0.251·63-s + 0.733·67-s − 1.44·69-s − 0.949·71-s − 1.17·73-s − 0.911·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−9.208724455562869122949802347769, −8.710604197360601493664037868588, −7.24131747550481313835329619950, −6.68069903466341434369024036297, −5.95306812196787019411326083898, −5.18076882477335398963883448479, −4.10049080223462139414710602745, −3.14183892066058120113423033146, −1.47887131331447322962254799675, 0,
1.47887131331447322962254799675, 3.14183892066058120113423033146, 4.10049080223462139414710602745, 5.18076882477335398963883448479, 5.95306812196787019411326083898, 6.68069903466341434369024036297, 7.24131747550481313835329619950, 8.710604197360601493664037868588, 9.208724455562869122949802347769
