| L(s) = 1 | + 5·7-s − 6·11-s − 3·13-s + 2·17-s − 19-s − 2·23-s − 6·29-s − 3·31-s − 6·37-s − 4·41-s − 11·43-s − 10·47-s + 18·49-s + 8·53-s − 6·59-s + 3·61-s + 67-s − 12·71-s + 10·73-s − 30·77-s + 8·79-s − 6·83-s + 16·89-s − 15·91-s − 7·97-s + 8·101-s − 4·103-s + ⋯ |
| L(s) = 1 | + 1.88·7-s − 1.80·11-s − 0.832·13-s + 0.485·17-s − 0.229·19-s − 0.417·23-s − 1.11·29-s − 0.538·31-s − 0.986·37-s − 0.624·41-s − 1.67·43-s − 1.45·47-s + 18/7·49-s + 1.09·53-s − 0.781·59-s + 0.384·61-s + 0.122·67-s − 1.42·71-s + 1.17·73-s − 3.41·77-s + 0.900·79-s − 0.658·83-s + 1.69·89-s − 1.57·91-s − 0.710·97-s + 0.796·101-s − 0.394·103-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 | \( 1 - 5 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 3 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 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
−8.023028787812861670563428883417, −7.70146752129486484186468370343, −6.91759127586569236268209773860, −5.53387390749850985695214123392, −5.19538963893584014035682043230, −4.60275522597127103441263563020, −3.43835761342060134820195762598, −2.30006608942557580680310669186, −1.67243127273087395873428033385, 0,
1.67243127273087395873428033385, 2.30006608942557580680310669186, 3.43835761342060134820195762598, 4.60275522597127103441263563020, 5.19538963893584014035682043230, 5.53387390749850985695214123392, 6.91759127586569236268209773860, 7.70146752129486484186468370343, 8.023028787812861670563428883417
