L(s) = 1 | − 3-s − 5·7-s + 9-s + 11-s − 3·13-s − 17-s + 5·19-s + 5·21-s − 4·23-s − 27-s + 7·31-s − 33-s + 2·37-s + 3·39-s + 43-s + 2·47-s + 18·49-s + 51-s + 12·53-s − 5·57-s + 12·59-s − 5·61-s − 5·63-s − 13·67-s + 4·69-s + 6·71-s − 10·73-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 1.88·7-s + 1/3·9-s + 0.301·11-s − 0.832·13-s − 0.242·17-s + 1.14·19-s + 1.09·21-s − 0.834·23-s − 0.192·27-s + 1.25·31-s − 0.174·33-s + 0.328·37-s + 0.480·39-s + 0.152·43-s + 0.291·47-s + 18/7·49-s + 0.140·51-s + 1.64·53-s − 0.662·57-s + 1.56·59-s − 0.640·61-s − 0.629·63-s − 1.58·67-s + 0.481·69-s + 0.712·71-s − 1.17·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 + T \) |
good | 7 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−13.18563103844394, −12.60132829234185, −12.27215334714011, −11.75942969214340, −11.63506818766031, −10.71327226502669, −10.29076180994735, −9.842873517199685, −9.684605433949437, −9.078665409922545, −8.615097170028022, −7.854107725263587, −7.327068305408419, −6.932777569113923, −6.517995665409003, −5.958712140546937, −5.639994381706063, −5.033803228208700, −4.290560146014061, −3.941398849668390, −3.281521235725495, −2.720605587943666, −2.322183683007843, −1.282233084869977, −0.6442618426547602, 0,
0.6442618426547602, 1.282233084869977, 2.322183683007843, 2.720605587943666, 3.281521235725495, 3.941398849668390, 4.290560146014061, 5.033803228208700, 5.639994381706063, 5.958712140546937, 6.517995665409003, 6.932777569113923, 7.327068305408419, 7.854107725263587, 8.615097170028022, 9.078665409922545, 9.684605433949437, 9.842873517199685, 10.29076180994735, 10.71327226502669, 11.63506818766031, 11.75942969214340, 12.27215334714011, 12.60132829234185, 13.18563103844394