| L(s) = 1 | + 3·5-s − 4·7-s − 3·17-s + 2·19-s + 6·23-s + 4·25-s − 9·29-s + 2·31-s − 12·35-s − 7·37-s − 3·41-s − 4·43-s + 6·47-s + 9·49-s − 9·53-s + 5·61-s + 2·67-s + 6·71-s − 73-s − 4·79-s − 12·83-s − 9·85-s − 6·89-s + 6·95-s + 14·97-s − 9·101-s + 2·103-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 1.51·7-s − 0.727·17-s + 0.458·19-s + 1.25·23-s + 4/5·25-s − 1.67·29-s + 0.359·31-s − 2.02·35-s − 1.15·37-s − 0.468·41-s − 0.609·43-s + 0.875·47-s + 9/7·49-s − 1.23·53-s + 0.640·61-s + 0.244·67-s + 0.712·71-s − 0.117·73-s − 0.450·79-s − 1.31·83-s − 0.976·85-s − 0.635·89-s + 0.615·95-s + 1.42·97-s − 0.895·101-s + 0.197·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6084 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6084 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.52407391257359586729652919556, −6.77251761069170647069609487137, −6.41277604194721839683308858573, −5.58682234214744189136920660436, −5.09522434006436411782676844639, −3.88477412167269184924387174177, −3.12139664768851773912143261726, −2.38420218495665159295499298451, −1.41869101434434330235627483491, 0,
1.41869101434434330235627483491, 2.38420218495665159295499298451, 3.12139664768851773912143261726, 3.88477412167269184924387174177, 5.09522434006436411782676844639, 5.58682234214744189136920660436, 6.41277604194721839683308858573, 6.77251761069170647069609487137, 7.52407391257359586729652919556
