| L(s) = 1 | − 5-s + 4·7-s − 11-s − 4·13-s − 6·17-s − 2·19-s + 6·23-s + 25-s − 6·29-s − 8·31-s − 4·35-s + 2·37-s − 6·41-s + 10·43-s + 6·47-s + 9·49-s + 6·53-s + 55-s + 8·61-s + 4·65-s + 4·67-s + 6·71-s + 14·73-s − 4·77-s + 16·79-s − 12·83-s + 6·85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.51·7-s − 0.301·11-s − 1.10·13-s − 1.45·17-s − 0.458·19-s + 1.25·23-s + 1/5·25-s − 1.11·29-s − 1.43·31-s − 0.676·35-s + 0.328·37-s − 0.937·41-s + 1.52·43-s + 0.875·47-s + 9/7·49-s + 0.824·53-s + 0.134·55-s + 1.02·61-s + 0.496·65-s + 0.488·67-s + 0.712·71-s + 1.63·73-s − 0.455·77-s + 1.80·79-s − 1.31·83-s + 0.650·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.719469832\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.719469832\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−7.64660009710299009584810308406, −7.37612371668070296964428912486, −6.64695441606065432452058471385, −5.49051200165531479837845000975, −5.02082294699543692595969424569, −4.40303777431817227556451692945, −3.67343424115108039790918121834, −2.39080333141427293328000383818, −1.98931130384689405849263868827, −0.63449038522764186602031158199,
0.63449038522764186602031158199, 1.98931130384689405849263868827, 2.39080333141427293328000383818, 3.67343424115108039790918121834, 4.40303777431817227556451692945, 5.02082294699543692595969424569, 5.49051200165531479837845000975, 6.64695441606065432452058471385, 7.37612371668070296964428912486, 7.64660009710299009584810308406
