| L(s) = 1 | + 3-s − 5-s − 2·9-s − 2·11-s + 4·13-s − 15-s − 6·19-s − 3·23-s + 25-s − 5·27-s + 3·29-s − 2·33-s + 12·37-s + 4·39-s + 7·41-s − 9·43-s + 2·45-s + 6·53-s + 2·55-s − 6·57-s + 10·59-s + 5·61-s − 4·65-s + 11·67-s − 3·69-s + 10·71-s + 8·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 2/3·9-s − 0.603·11-s + 1.10·13-s − 0.258·15-s − 1.37·19-s − 0.625·23-s + 1/5·25-s − 0.962·27-s + 0.557·29-s − 0.348·33-s + 1.97·37-s + 0.640·39-s + 1.09·41-s − 1.37·43-s + 0.298·45-s + 0.824·53-s + 0.269·55-s − 0.794·57-s + 1.30·59-s + 0.640·61-s − 0.496·65-s + 1.34·67-s − 0.361·69-s + 1.18·71-s + 0.936·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 12 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + 17 T + p T^{2} \) |
| 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
−16.19254086561140, −15.71073991763573, −15.03317096382938, −14.73355162016819, −14.06234899553830, −13.50481733326951, −12.99159172558534, −12.50948377010192, −11.65391400240433, −11.19880446655938, −10.76517241086868, −9.995941389378012, −9.401452928245634, −8.614459310488077, −8.159900855933899, −8.024748840777779, −6.947364092189233, −6.347507731376195, −5.738792646876990, −5.010811781724806, −3.968704186183275, −3.822755626043629, −2.658573554982230, −2.354943130528988, −1.100792536075777, 0,
1.100792536075777, 2.354943130528988, 2.658573554982230, 3.822755626043629, 3.968704186183275, 5.010811781724806, 5.738792646876990, 6.347507731376195, 6.947364092189233, 8.024748840777779, 8.159900855933899, 8.614459310488077, 9.401452928245634, 9.995941389378012, 10.76517241086868, 11.19880446655938, 11.65391400240433, 12.50948377010192, 12.99159172558534, 13.50481733326951, 14.06234899553830, 14.73355162016819, 15.03317096382938, 15.71073991763573, 16.19254086561140
