| L(s) = 1 | + 2-s + 4-s − 7-s + 8-s − 3·11-s − 2·13-s − 14-s + 16-s + 3·17-s − 7·19-s − 3·22-s − 2·26-s − 28-s + 6·29-s − 4·31-s + 32-s + 3·34-s − 8·37-s − 7·38-s + 9·41-s − 8·43-s − 3·44-s − 6·47-s + 49-s − 2·52-s − 12·53-s − 56-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s − 0.904·11-s − 0.554·13-s − 0.267·14-s + 1/4·16-s + 0.727·17-s − 1.60·19-s − 0.639·22-s − 0.392·26-s − 0.188·28-s + 1.11·29-s − 0.718·31-s + 0.176·32-s + 0.514·34-s − 1.31·37-s − 1.13·38-s + 1.40·41-s − 1.21·43-s − 0.452·44-s − 0.875·47-s + 1/7·49-s − 0.277·52-s − 1.64·53-s − 0.133·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−8.128175988279174688342998749037, −7.54916562418446708067081633136, −6.61655337084746600555916250971, −6.06041242149753935532362309037, −5.09749037578595124394213303559, −4.55717585193744392972685513969, −3.47248853790241650296879805850, −2.74930206746252115572407189719, −1.75943695494265392225703556586, 0,
1.75943695494265392225703556586, 2.74930206746252115572407189719, 3.47248853790241650296879805850, 4.55717585193744392972685513969, 5.09749037578595124394213303559, 6.06041242149753935532362309037, 6.61655337084746600555916250971, 7.54916562418446708067081633136, 8.128175988279174688342998749037
