| L(s) = 1 | + 2·3-s − 5-s − 2·7-s + 9-s + 13-s − 2·15-s + 5·17-s − 6·19-s − 4·21-s − 2·23-s − 4·25-s − 4·27-s + 9·29-s + 2·31-s + 2·35-s + 3·37-s + 2·39-s + 5·41-s − 45-s − 2·47-s − 3·49-s + 10·51-s − 9·53-s − 12·57-s + 8·59-s + 6·61-s − 2·63-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s + 0.277·13-s − 0.516·15-s + 1.21·17-s − 1.37·19-s − 0.872·21-s − 0.417·23-s − 4/5·25-s − 0.769·27-s + 1.67·29-s + 0.359·31-s + 0.338·35-s + 0.493·37-s + 0.320·39-s + 0.780·41-s − 0.149·45-s − 0.291·47-s − 3/7·49-s + 1.40·51-s − 1.23·53-s − 1.58·57-s + 1.04·59-s + 0.768·61-s − 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−7.81304245413044943247521419550, −6.86022704125109365573887961245, −6.23631640550960698833745661801, −5.51160152763630926450656211881, −4.31690254498802738209818156699, −3.87299050839595889181889379856, −3.01368167787977742950054040757, −2.55123860245483446996298704576, −1.38763061579404254635807455108, 0,
1.38763061579404254635807455108, 2.55123860245483446996298704576, 3.01368167787977742950054040757, 3.87299050839595889181889379856, 4.31690254498802738209818156699, 5.51160152763630926450656211881, 6.23631640550960698833745661801, 6.86022704125109365573887961245, 7.81304245413044943247521419550
