| L(s) = 1 | − 2·5-s − 2·7-s − 2·11-s − 13-s − 6·17-s − 2·19-s − 25-s − 2·29-s + 2·31-s + 4·35-s + 10·37-s − 2·41-s + 8·43-s + 2·47-s − 3·49-s + 2·53-s + 4·55-s − 10·59-s + 6·61-s + 2·65-s + 14·67-s + 14·71-s − 6·73-s + 4·77-s − 4·79-s − 6·83-s + 12·85-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.755·7-s − 0.603·11-s − 0.277·13-s − 1.45·17-s − 0.458·19-s − 1/5·25-s − 0.371·29-s + 0.359·31-s + 0.676·35-s + 1.64·37-s − 0.312·41-s + 1.21·43-s + 0.291·47-s − 3/7·49-s + 0.274·53-s + 0.539·55-s − 1.30·59-s + 0.768·61-s + 0.248·65-s + 1.71·67-s + 1.66·71-s − 0.702·73-s + 0.455·77-s − 0.450·79-s − 0.658·83-s + 1.30·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7510882941\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7510882941\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−8.378388028011655328087417015303, −7.84277873015020165975111568842, −7.03211669026227018049059408323, −6.41075448523315206628985215899, −5.55718106230353369331500920139, −4.49375935423844689453310654652, −4.01111060168236157128614254869, −2.96712380651428298766874018198, −2.18571735072400801587493852448, −0.47448241934470296459246464935,
0.47448241934470296459246464935, 2.18571735072400801587493852448, 2.96712380651428298766874018198, 4.01111060168236157128614254869, 4.49375935423844689453310654652, 5.55718106230353369331500920139, 6.41075448523315206628985215899, 7.03211669026227018049059408323, 7.84277873015020165975111568842, 8.378388028011655328087417015303
