L(s) = 1 | − 6·13-s − 2·17-s − 8·19-s − 8·23-s + 2·29-s + 4·31-s + 2·37-s − 6·41-s + 4·43-s + 8·47-s + 10·53-s − 4·59-s + 2·61-s + 4·67-s − 12·71-s − 2·73-s − 8·79-s − 4·83-s − 6·89-s − 18·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
L(s) = 1 | − 1.66·13-s − 0.485·17-s − 1.83·19-s − 1.66·23-s + 0.371·29-s + 0.718·31-s + 0.328·37-s − 0.937·41-s + 0.609·43-s + 1.16·47-s + 1.37·53-s − 0.520·59-s + 0.256·61-s + 0.488·67-s − 1.42·71-s − 0.234·73-s − 0.900·79-s − 0.439·83-s − 0.635·89-s − 1.82·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−13.67346843961417, −13.17693758324475, −12.58153813909968, −12.24837390209479, −11.91363096314277, −11.34861732103148, −10.71667645902272, −10.24475309912066, −10.00743338222712, −9.450801990624473, −8.747374463524334, −8.488519166928601, −7.880361096255469, −7.411374929703662, −6.836598055696052, −6.439033325130713, −5.845471756201932, −5.346325400956165, −4.659968196458529, −4.156372566304164, −3.983171754618743, −2.819095466228097, −2.479014035366735, −2.036028346215775, −1.214946165904799, 0, 0,
1.214946165904799, 2.036028346215775, 2.479014035366735, 2.819095466228097, 3.983171754618743, 4.156372566304164, 4.659968196458529, 5.346325400956165, 5.845471756201932, 6.439033325130713, 6.836598055696052, 7.411374929703662, 7.880361096255469, 8.488519166928601, 8.747374463524334, 9.450801990624473, 10.00743338222712, 10.24475309912066, 10.71667645902272, 11.34861732103148, 11.91363096314277, 12.24837390209479, 12.58153813909968, 13.17693758324475, 13.67346843961417