L(s) = 1 | − 3-s + 1.73·7-s + 9-s + 1.73·11-s + 13-s − 2.46·17-s − 3.46·19-s − 1.73·21-s + 2·23-s − 27-s + 3.92·29-s − 9.19·31-s − 1.73·33-s − 0.535·37-s − 39-s − 2.53·41-s + 8.92·43-s − 6.66·47-s − 4·49-s + 2.46·51-s − 11.9·53-s + 3.46·57-s + 1.73·59-s − 12.4·61-s + 1.73·63-s + 4.26·67-s − 2·69-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.654·7-s + 0.333·9-s + 0.522·11-s + 0.277·13-s − 0.597·17-s − 0.794·19-s − 0.377·21-s + 0.417·23-s − 0.192·27-s + 0.729·29-s − 1.65·31-s − 0.301·33-s − 0.0881·37-s − 0.160·39-s − 0.396·41-s + 1.36·43-s − 0.971·47-s − 0.571·49-s + 0.345·51-s − 1.63·53-s + 0.458·57-s + 0.225·59-s − 1.59·61-s + 0.218·63-s + 0.521·67-s − 0.240·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 - 1.73T + 7T^{2} \) |
| 11 | \( 1 - 1.73T + 11T^{2} \) |
| 17 | \( 1 + 2.46T + 17T^{2} \) |
| 19 | \( 1 + 3.46T + 19T^{2} \) |
| 23 | \( 1 - 2T + 23T^{2} \) |
| 29 | \( 1 - 3.92T + 29T^{2} \) |
| 31 | \( 1 + 9.19T + 31T^{2} \) |
| 37 | \( 1 + 0.535T + 37T^{2} \) |
| 41 | \( 1 + 2.53T + 41T^{2} \) |
| 43 | \( 1 - 8.92T + 43T^{2} \) |
| 47 | \( 1 + 6.66T + 47T^{2} \) |
| 53 | \( 1 + 11.9T + 53T^{2} \) |
| 59 | \( 1 - 1.73T + 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 - 4.26T + 67T^{2} \) |
| 71 | \( 1 + 14.3T + 71T^{2} \) |
| 73 | \( 1 + 3.46T + 73T^{2} \) |
| 79 | \( 1 - 14T + 79T^{2} \) |
| 83 | \( 1 - 13.7T + 83T^{2} \) |
| 89 | \( 1 + 2.92T + 89T^{2} \) |
| 97 | \( 1 - 4.92T + 97T^{2} \) |
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
−7.52846404653819358300152975543, −6.62905062643755033230210897924, −6.26769544841163805648943098354, −5.35686278140064492046068272778, −4.70025430564873472196244825207, −4.09700886591462600377067140390, −3.17881188964903888592991475725, −2.02799230339822298338267029124, −1.30774773316930170918581049325, 0,
1.30774773316930170918581049325, 2.02799230339822298338267029124, 3.17881188964903888592991475725, 4.09700886591462600377067140390, 4.70025430564873472196244825207, 5.35686278140064492046068272778, 6.26769544841163805648943098354, 6.62905062643755033230210897924, 7.52846404653819358300152975543