| L(s) = 1 | − 5-s + 2·7-s − 2·13-s − 4·17-s + 19-s − 4·23-s + 25-s + 6·29-s + 10·31-s − 2·35-s − 10·37-s + 4·43-s − 4·47-s − 3·49-s − 10·53-s + 6·59-s + 10·61-s + 2·65-s − 4·67-s − 8·71-s − 2·73-s − 6·79-s − 6·83-s + 4·85-s − 4·89-s − 4·91-s − 95-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.755·7-s − 0.554·13-s − 0.970·17-s + 0.229·19-s − 0.834·23-s + 1/5·25-s + 1.11·29-s + 1.79·31-s − 0.338·35-s − 1.64·37-s + 0.609·43-s − 0.583·47-s − 3/7·49-s − 1.37·53-s + 0.781·59-s + 1.28·61-s + 0.248·65-s − 0.488·67-s − 0.949·71-s − 0.234·73-s − 0.675·79-s − 0.658·83-s + 0.433·85-s − 0.423·89-s − 0.419·91-s − 0.102·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 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
−7.72368761561864544690209103113, −6.89606727991333534015960401115, −6.36705302215135246401202472023, −5.34831909124529015765270120238, −4.66652388752455122764586993329, −4.18585384563425980316495790821, −3.10948701275610497190123065064, −2.30099037886864045378327535292, −1.30775419494860434547175997464, 0,
1.30775419494860434547175997464, 2.30099037886864045378327535292, 3.10948701275610497190123065064, 4.18585384563425980316495790821, 4.66652388752455122764586993329, 5.34831909124529015765270120238, 6.36705302215135246401202472023, 6.89606727991333534015960401115, 7.72368761561864544690209103113
