| L(s) = 1 | + 3-s − 3·5-s + 7-s + 9-s − 13-s − 3·15-s − 4·17-s + 21-s − 23-s + 4·25-s + 27-s + 5·29-s + 10·31-s − 3·35-s − 3·37-s − 39-s − 11·41-s + 9·43-s − 3·45-s + 3·47-s + 49-s − 4·51-s + 12·53-s − 10·59-s − 10·61-s + 63-s + 3·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.34·5-s + 0.377·7-s + 1/3·9-s − 0.277·13-s − 0.774·15-s − 0.970·17-s + 0.218·21-s − 0.208·23-s + 4/5·25-s + 0.192·27-s + 0.928·29-s + 1.79·31-s − 0.507·35-s − 0.493·37-s − 0.160·39-s − 1.71·41-s + 1.37·43-s − 0.447·45-s + 0.437·47-s + 1/7·49-s − 0.560·51-s + 1.64·53-s − 1.30·59-s − 1.28·61-s + 0.125·63-s + 0.372·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 11 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 9 T + p T^{2} \) |
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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
−7.59965267583394120895466385026, −7.00217750434419845410448942993, −6.33582082496062884460268580844, −5.21569438121451725456142195156, −4.38789318063603582788983743599, −4.10721408376758978447161596513, −3.08400932814634790744861853224, −2.43378750100698839758206508134, −1.23347331354479212409499920845, 0,
1.23347331354479212409499920845, 2.43378750100698839758206508134, 3.08400932814634790744861853224, 4.10721408376758978447161596513, 4.38789318063603582788983743599, 5.21569438121451725456142195156, 6.33582082496062884460268580844, 7.00217750434419845410448942993, 7.59965267583394120895466385026
