L(s) = 1 | + 3-s − 2·9-s + 6·11-s − 3·13-s − 17-s + 7·19-s − 8·23-s − 5·27-s − 5·29-s − 5·31-s + 6·33-s − 8·37-s − 3·39-s − 4·43-s + 3·47-s − 7·49-s − 51-s − 9·53-s + 7·57-s − 5·59-s − 3·61-s − 2·67-s − 8·69-s + 15·71-s + 11·73-s − 8·79-s + 81-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 2/3·9-s + 1.80·11-s − 0.832·13-s − 0.242·17-s + 1.60·19-s − 1.66·23-s − 0.962·27-s − 0.928·29-s − 0.898·31-s + 1.04·33-s − 1.31·37-s − 0.480·39-s − 0.609·43-s + 0.437·47-s − 49-s − 0.140·51-s − 1.23·53-s + 0.927·57-s − 0.650·59-s − 0.384·61-s − 0.244·67-s − 0.963·69-s + 1.78·71-s + 1.28·73-s − 0.900·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 17 | \( 1 + T \) |
good | 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 + 3 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 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
−7.67190967407261500715487657327, −7.00456867547223699734923627671, −6.23985496076442958715764471221, −5.54071766389077854290634975130, −4.73742237198230170422127614987, −3.64876638837624385308800889734, −3.43456356948315906377922497380, −2.21120751918900987379160924997, −1.50276255157466343678431379749, 0,
1.50276255157466343678431379749, 2.21120751918900987379160924997, 3.43456356948315906377922497380, 3.64876638837624385308800889734, 4.73742237198230170422127614987, 5.54071766389077854290634975130, 6.23985496076442958715764471221, 7.00456867547223699734923627671, 7.67190967407261500715487657327