| L(s) = 1 | + 3-s + 5-s − 4·7-s + 9-s − 3·13-s + 15-s − 17-s + 8·19-s − 4·21-s − 8·23-s − 4·25-s + 27-s + 9·29-s − 4·35-s − 3·37-s − 3·39-s + 3·41-s + 8·43-s + 45-s + 12·47-s + 9·49-s − 51-s − 11·53-s + 8·57-s + 2·61-s − 4·63-s − 3·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 1.51·7-s + 1/3·9-s − 0.832·13-s + 0.258·15-s − 0.242·17-s + 1.83·19-s − 0.872·21-s − 1.66·23-s − 4/5·25-s + 0.192·27-s + 1.67·29-s − 0.676·35-s − 0.493·37-s − 0.480·39-s + 0.468·41-s + 1.21·43-s + 0.149·45-s + 1.75·47-s + 9/7·49-s − 0.140·51-s − 1.51·53-s + 1.05·57-s + 0.256·61-s − 0.503·63-s − 0.372·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−15.66718283894187, −15.55178656540856, −14.33883140593426, −14.05937198185969, −13.76890932334475, −13.08840714081018, −12.43315403323712, −12.15792967107145, −11.54021327719655, −10.54066435269681, −9.950596051493226, −9.809085760213519, −9.243786913758720, −8.658283664179223, −7.773783307941537, −7.394548704340756, −6.755522830854376, −5.979969453133889, −5.720940738813309, −4.702263698838742, −4.041530471475728, −3.295284708960137, −2.752355087799618, −2.159186284457880, −1.067614970092562, 0,
1.067614970092562, 2.159186284457880, 2.752355087799618, 3.295284708960137, 4.041530471475728, 4.702263698838742, 5.720940738813309, 5.979969453133889, 6.755522830854376, 7.394548704340756, 7.773783307941537, 8.658283664179223, 9.243786913758720, 9.809085760213519, 9.950596051493226, 10.54066435269681, 11.54021327719655, 12.15792967107145, 12.43315403323712, 13.08840714081018, 13.76890932334475, 14.05937198185969, 14.33883140593426, 15.55178656540856, 15.66718283894187
