| L(s) = 1 | + 3·3-s + 2·5-s − 3·7-s + 6·9-s + 3·11-s − 5·13-s + 6·15-s − 8·17-s + 4·19-s − 9·21-s + 5·23-s − 25-s + 9·27-s + 2·31-s + 9·33-s − 6·35-s − 4·37-s − 15·39-s − 10·41-s − 43-s + 12·45-s + 3·47-s + 2·49-s − 24·51-s + 12·53-s + 6·55-s + 12·57-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.894·5-s − 1.13·7-s + 2·9-s + 0.904·11-s − 1.38·13-s + 1.54·15-s − 1.94·17-s + 0.917·19-s − 1.96·21-s + 1.04·23-s − 1/5·25-s + 1.73·27-s + 0.359·31-s + 1.56·33-s − 1.01·35-s − 0.657·37-s − 2.40·39-s − 1.56·41-s − 0.152·43-s + 1.78·45-s + 0.437·47-s + 2/7·49-s − 3.36·51-s + 1.64·53-s + 0.809·55-s + 1.58·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.737645376\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.737645376\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 503 | \( 1 + T \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−14.96115650393445, −14.52607290067417, −13.84239670073316, −13.59295666117853, −13.14075921822389, −12.72594966607296, −11.96655488263871, −11.43658096422267, −10.40411018969466, −9.951715892155908, −9.600874147173894, −9.102791594387554, −8.804956710848001, −8.145396609920112, −7.227983770373877, −6.785209606769515, −6.660522880960148, −5.511670327304081, −4.895640659355626, −4.079130129590736, −3.527734812256464, −2.881142228309982, −2.253021437339602, −1.913941241787505, −0.7110802732657953,
0.7110802732657953, 1.913941241787505, 2.253021437339602, 2.881142228309982, 3.527734812256464, 4.079130129590736, 4.895640659355626, 5.511670327304081, 6.660522880960148, 6.785209606769515, 7.227983770373877, 8.145396609920112, 8.804956710848001, 9.102791594387554, 9.600874147173894, 9.951715892155908, 10.40411018969466, 11.43658096422267, 11.96655488263871, 12.72594966607296, 13.14075921822389, 13.59295666117853, 13.84239670073316, 14.52607290067417, 14.96115650393445
