| L(s) = 1 | + 3-s − 5-s + 4·7-s + 9-s − 3·11-s − 13-s − 15-s − 17-s − 19-s + 4·21-s + 6·23-s + 25-s + 27-s − 5·31-s − 3·33-s − 4·35-s − 11·37-s − 39-s − 6·41-s + 5·43-s − 45-s + 12·47-s + 9·49-s − 51-s + 12·53-s + 3·55-s − 57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1.51·7-s + 1/3·9-s − 0.904·11-s − 0.277·13-s − 0.258·15-s − 0.242·17-s − 0.229·19-s + 0.872·21-s + 1.25·23-s + 1/5·25-s + 0.192·27-s − 0.898·31-s − 0.522·33-s − 0.676·35-s − 1.80·37-s − 0.160·39-s − 0.937·41-s + 0.762·43-s − 0.149·45-s + 1.75·47-s + 9/7·49-s − 0.140·51-s + 1.64·53-s + 0.404·55-s − 0.132·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 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
−13.31796467989237, −12.76452797935895, −12.24248776255035, −11.93781321361721, −11.29767028650815, −10.81277827241128, −10.56821233500040, −10.15051573753251, −9.218923613792532, −8.883907531798803, −8.583576622014462, −8.014481756499246, −7.480109552454182, −7.304620877333724, −6.768708254012831, −5.863513454769558, −5.273146563327234, −5.025360374941143, −4.420458691189597, −3.960138480706673, −3.266846475149994, −2.713945576781757, −2.109422673946741, −1.622496864687315, −0.8895313134922975, 0,
0.8895313134922975, 1.622496864687315, 2.109422673946741, 2.713945576781757, 3.266846475149994, 3.960138480706673, 4.420458691189597, 5.025360374941143, 5.273146563327234, 5.863513454769558, 6.768708254012831, 7.304620877333724, 7.480109552454182, 8.014481756499246, 8.583576622014462, 8.883907531798803, 9.218923613792532, 10.15051573753251, 10.56821233500040, 10.81277827241128, 11.29767028650815, 11.93781321361721, 12.24248776255035, 12.76452797935895, 13.31796467989237
