L(s) = 1 | − 3-s − 4·7-s + 9-s + 4·11-s + 13-s − 2·17-s + 8·19-s + 4·21-s − 27-s − 6·29-s − 4·31-s − 4·33-s − 2·37-s − 39-s − 10·41-s + 4·43-s − 8·47-s + 9·49-s + 2·51-s − 10·53-s − 8·57-s − 4·59-s + 2·61-s − 4·63-s − 16·67-s − 8·71-s − 2·73-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 1.51·7-s + 1/3·9-s + 1.20·11-s + 0.277·13-s − 0.485·17-s + 1.83·19-s + 0.872·21-s − 0.192·27-s − 1.11·29-s − 0.718·31-s − 0.696·33-s − 0.328·37-s − 0.160·39-s − 1.56·41-s + 0.609·43-s − 1.16·47-s + 9/7·49-s + 0.280·51-s − 1.37·53-s − 1.05·57-s − 0.520·59-s + 0.256·61-s − 0.503·63-s − 1.95·67-s − 0.949·71-s − 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9173362253\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9173362253\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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
−14.16107788764010, −13.64925029106551, −13.24377135407827, −12.79962672404359, −12.07041053327919, −11.89863895327777, −11.28608460553366, −10.76269459676330, −10.11710197410330, −9.603239753904506, −9.224014047954817, −8.923602017513256, −7.926903507613855, −7.345747239349982, −6.855962767345110, −6.383866382528157, −5.952253267382424, −5.339977394325979, −4.702104913499659, −3.910378181355136, −3.366109793485534, −3.089498074718448, −1.875428993749397, −1.305212843070448, −0.3513693356622415,
0.3513693356622415, 1.305212843070448, 1.875428993749397, 3.089498074718448, 3.366109793485534, 3.910378181355136, 4.702104913499659, 5.339977394325979, 5.952253267382424, 6.383866382528157, 6.855962767345110, 7.345747239349982, 7.926903507613855, 8.923602017513256, 9.224014047954817, 9.603239753904506, 10.11710197410330, 10.76269459676330, 11.28608460553366, 11.89863895327777, 12.07041053327919, 12.79962672404359, 13.24377135407827, 13.64925029106551, 14.16107788764010