L(s) = 1 | + 5-s − 7-s − 4·13-s − 6·17-s + 2·19-s − 3·23-s + 25-s + 3·29-s − 10·31-s − 35-s − 10·37-s + 9·41-s − 4·43-s + 9·47-s − 6·49-s − 6·53-s − 6·59-s − 61-s − 4·65-s + 11·67-s + 12·71-s − 4·73-s − 10·79-s − 9·83-s − 6·85-s + 9·89-s + 4·91-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 0.377·7-s − 1.10·13-s − 1.45·17-s + 0.458·19-s − 0.625·23-s + 1/5·25-s + 0.557·29-s − 1.79·31-s − 0.169·35-s − 1.64·37-s + 1.40·41-s − 0.609·43-s + 1.31·47-s − 6/7·49-s − 0.824·53-s − 0.781·59-s − 0.128·61-s − 0.496·65-s + 1.34·67-s + 1.42·71-s − 0.468·73-s − 1.12·79-s − 0.987·83-s − 0.650·85-s + 0.953·89-s + 0.419·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−9.211557631327800264226337570974, −8.250652739403387235530819275152, −7.24684325165229751785808403176, −6.68154082547930757538776222292, −5.69579728838676039234716015879, −4.90626234426666430241110404100, −3.92321978955800419934558403450, −2.75239535672486773062571767044, −1.84236295216713954023001163171, 0,
1.84236295216713954023001163171, 2.75239535672486773062571767044, 3.92321978955800419934558403450, 4.90626234426666430241110404100, 5.69579728838676039234716015879, 6.68154082547930757538776222292, 7.24684325165229751785808403176, 8.250652739403387235530819275152, 9.211557631327800264226337570974