L(s) = 1 | + 3-s + 5-s + 9-s + 11-s + 4·13-s + 15-s + 2·17-s − 2·19-s − 8·23-s + 25-s + 27-s − 4·29-s + 4·31-s + 33-s − 2·37-s + 4·39-s + 12·41-s + 8·43-s + 45-s + 4·47-s + 2·51-s − 10·53-s + 55-s − 2·57-s − 4·59-s − 6·61-s + 4·65-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1/3·9-s + 0.301·11-s + 1.10·13-s + 0.258·15-s + 0.485·17-s − 0.458·19-s − 1.66·23-s + 1/5·25-s + 0.192·27-s − 0.742·29-s + 0.718·31-s + 0.174·33-s − 0.328·37-s + 0.640·39-s + 1.87·41-s + 1.21·43-s + 0.149·45-s + 0.583·47-s + 0.280·51-s − 1.37·53-s + 0.134·55-s − 0.264·57-s − 0.520·59-s − 0.768·61-s + 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.664973639\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.664973639\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−15.09219448559995, −14.21884598800323, −14.03875561777932, −13.74656490693546, −12.79491549557895, −12.61438885536323, −11.98591396448958, −11.07930887076628, −10.92122444678182, −10.12032652319635, −9.593553049026740, −9.181790178577783, −8.547836002963537, −7.955614741488484, −7.606024841891754, −6.705901301758420, −6.092657073280834, −5.835109646672275, −4.918979159808012, −4.038715329718409, −3.831737812289306, −2.917153710261776, −2.210832831511986, −1.567395211760324, −0.7167267793461253,
0.7167267793461253, 1.567395211760324, 2.210832831511986, 2.917153710261776, 3.831737812289306, 4.038715329718409, 4.918979159808012, 5.835109646672275, 6.092657073280834, 6.705901301758420, 7.606024841891754, 7.955614741488484, 8.547836002963537, 9.181790178577783, 9.593553049026740, 10.12032652319635, 10.92122444678182, 11.07930887076628, 11.98591396448958, 12.61438885536323, 12.79491549557895, 13.74656490693546, 14.03875561777932, 14.21884598800323, 15.09219448559995