L(s) = 1 | + 4·5-s − 2·7-s + 11-s + 2·13-s + 2·17-s + 6·19-s + 4·23-s + 11·25-s + 6·29-s + 4·31-s − 8·35-s + 6·37-s − 10·41-s − 6·43-s − 8·47-s − 3·49-s + 4·55-s − 4·59-s + 6·61-s + 8·65-s − 8·67-s − 2·73-s − 2·77-s − 10·79-s − 12·83-s + 8·85-s − 4·91-s + ⋯ |
L(s) = 1 | + 1.78·5-s − 0.755·7-s + 0.301·11-s + 0.554·13-s + 0.485·17-s + 1.37·19-s + 0.834·23-s + 11/5·25-s + 1.11·29-s + 0.718·31-s − 1.35·35-s + 0.986·37-s − 1.56·41-s − 0.914·43-s − 1.16·47-s − 3/7·49-s + 0.539·55-s − 0.520·59-s + 0.768·61-s + 0.992·65-s − 0.977·67-s − 0.234·73-s − 0.227·77-s − 1.12·79-s − 1.31·83-s + 0.867·85-s − 0.419·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.180886540\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.180886540\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−8.161397174305910128840219629648, −7.04946672754783872465527385107, −6.52796407082151734123033810027, −5.98287642672534840523826158016, −5.30162414209343187590008792160, −4.63088121216365101084881746326, −3.21081016263465542739759135936, −2.95968678937684726040166683541, −1.71232536509978533851290479437, −1.01169248374816977272370570345,
1.01169248374816977272370570345, 1.71232536509978533851290479437, 2.95968678937684726040166683541, 3.21081016263465542739759135936, 4.63088121216365101084881746326, 5.30162414209343187590008792160, 5.98287642672534840523826158016, 6.52796407082151734123033810027, 7.04946672754783872465527385107, 8.161397174305910128840219629648