L(s) = 1 | − 3-s − 7-s + 9-s + 11-s + 2·13-s − 2·19-s + 21-s − 5·25-s − 27-s − 6·29-s − 2·31-s − 33-s + 2·37-s − 2·39-s + 4·43-s + 6·47-s + 49-s − 6·53-s + 2·57-s + 2·61-s − 63-s + 4·67-s + 12·71-s − 4·73-s + 5·75-s − 77-s − 8·79-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.301·11-s + 0.554·13-s − 0.458·19-s + 0.218·21-s − 25-s − 0.192·27-s − 1.11·29-s − 0.359·31-s − 0.174·33-s + 0.328·37-s − 0.320·39-s + 0.609·43-s + 0.875·47-s + 1/7·49-s − 0.824·53-s + 0.264·57-s + 0.256·61-s − 0.125·63-s + 0.488·67-s + 1.42·71-s − 0.468·73-s + 0.577·75-s − 0.113·77-s − 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3696 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
good | 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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
−8.091864802864548946274201238617, −7.36296884237547206221182696584, −6.56882992459545583820896267717, −5.93513117193887118738043043803, −5.30391829089435212285661993165, −4.18660148985390784722888338849, −3.67589106183491380823479177230, −2.43761285337526943822699134915, −1.34866118796415812108005446138, 0,
1.34866118796415812108005446138, 2.43761285337526943822699134915, 3.67589106183491380823479177230, 4.18660148985390784722888338849, 5.30391829089435212285661993165, 5.93513117193887118738043043803, 6.56882992459545583820896267717, 7.36296884237547206221182696584, 8.091864802864548946274201238617