| L(s) = 1 | − 4·5-s + 4·7-s − 13-s + 17-s − 6·19-s + 11·25-s + 2·29-s − 16·35-s − 4·37-s − 4·43-s − 6·47-s + 9·49-s + 2·53-s + 6·59-s − 2·61-s + 4·65-s − 14·67-s + 12·71-s − 4·73-s + 6·83-s − 4·85-s + 18·89-s − 4·91-s + 24·95-s − 16·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 1.51·7-s − 0.277·13-s + 0.242·17-s − 1.37·19-s + 11/5·25-s + 0.371·29-s − 2.70·35-s − 0.657·37-s − 0.609·43-s − 0.875·47-s + 9/7·49-s + 0.274·53-s + 0.781·59-s − 0.256·61-s + 0.496·65-s − 1.71·67-s + 1.42·71-s − 0.468·73-s + 0.658·83-s − 0.433·85-s + 1.90·89-s − 0.419·91-s + 2.46·95-s − 1.62·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 127296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 127296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−13.75560049292588, −13.27478009229788, −12.56137246739908, −12.16246928574742, −11.87295405007364, −11.31020902739555, −11.00474715673809, −10.54506426668517, −10.05012357332526, −9.154899569112182, −8.612831835758109, −8.319517157029156, −7.882134743847756, −7.529320206515474, −6.883718070266172, −6.483199814876884, −5.566189156673448, −4.989842687249510, −4.562914572214506, −4.185333283350187, −3.594158941622210, −2.990116694786704, −2.174989827230201, −1.569985915877883, −0.7465566815042748, 0,
0.7465566815042748, 1.569985915877883, 2.174989827230201, 2.990116694786704, 3.594158941622210, 4.185333283350187, 4.562914572214506, 4.989842687249510, 5.566189156673448, 6.483199814876884, 6.883718070266172, 7.529320206515474, 7.882134743847756, 8.319517157029156, 8.612831835758109, 9.154899569112182, 10.05012357332526, 10.54506426668517, 11.00474715673809, 11.31020902739555, 11.87295405007364, 12.16246928574742, 12.56137246739908, 13.27478009229788, 13.75560049292588
