| L(s) = 1 | − 3·7-s + 2·11-s − 3·13-s − 6·17-s + 7·19-s + 6·23-s + 2·29-s + 5·31-s + 10·37-s − 12·41-s − 3·43-s − 10·47-s + 2·49-s − 6·59-s − 13·61-s − 7·67-s − 4·71-s − 6·73-s − 6·77-s + 8·79-s − 6·83-s − 16·89-s + 9·91-s − 7·97-s + 12·103-s − 16·107-s + 9·109-s + ⋯ |
| L(s) = 1 | − 1.13·7-s + 0.603·11-s − 0.832·13-s − 1.45·17-s + 1.60·19-s + 1.25·23-s + 0.371·29-s + 0.898·31-s + 1.64·37-s − 1.87·41-s − 0.457·43-s − 1.45·47-s + 2/7·49-s − 0.781·59-s − 1.66·61-s − 0.855·67-s − 0.474·71-s − 0.702·73-s − 0.683·77-s + 0.900·79-s − 0.658·83-s − 1.69·89-s + 0.943·91-s − 0.710·97-s + 1.18·103-s − 1.54·107-s + 0.862·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.217130195737089314643827465817, −7.25402032673487199321934238648, −6.71649189153655010536111525845, −6.13603427500219351094103491808, −5.02564735992692304106368945332, −4.43513048617502010957575901259, −3.22800736564922508959940330825, −2.79002466592839737156701580419, −1.37918655758222598324442176639, 0,
1.37918655758222598324442176639, 2.79002466592839737156701580419, 3.22800736564922508959940330825, 4.43513048617502010957575901259, 5.02564735992692304106368945332, 6.13603427500219351094103491808, 6.71649189153655010536111525845, 7.25402032673487199321934238648, 8.217130195737089314643827465817
