| L(s) = 1 | + 2-s − 4-s − 5-s − 3·8-s − 10-s − 4·13-s − 16-s + 20-s + 25-s − 4·26-s + 5·32-s − 20·37-s + 3·40-s − 20·41-s + 8·43-s − 14·49-s + 50-s + 4·52-s + 20·53-s + 7·64-s + 4·65-s + 24·67-s + 16·71-s − 20·74-s + 80-s − 20·82-s − 24·83-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.447·5-s − 1.06·8-s − 0.316·10-s − 1.10·13-s − 1/4·16-s + 0.223·20-s + 1/5·25-s − 0.784·26-s + 0.883·32-s − 3.28·37-s + 0.474·40-s − 3.12·41-s + 1.21·43-s − 2·49-s + 0.141·50-s + 0.554·52-s + 2.74·53-s + 7/8·64-s + 0.496·65-s + 2.93·67-s + 1.89·71-s − 2.32·74-s + 0.111·80-s − 2.20·82-s − 2.63·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.074326590\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.074326590\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 - T + p T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( 1 + T \) |
| good | 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.528796002193558283113883461082, −8.052642541080546717924291648317, −7.30265771947486168528941346448, −6.87126538623379444026486653759, −6.76955885158912340897756950945, −5.96744368889847453387068579979, −5.32790836897096555983732668084, −5.04860090093668311608273372623, −4.86275236111181745763302133865, −3.97486198573635904909121919320, −3.63412818236731287070311550143, −3.23564539976077866970720545176, −2.43462761639899639642195755230, −1.76953374967632241466455046617, −0.45394994076227052108962099324,
0.45394994076227052108962099324, 1.76953374967632241466455046617, 2.43462761639899639642195755230, 3.23564539976077866970720545176, 3.63412818236731287070311550143, 3.97486198573635904909121919320, 4.86275236111181745763302133865, 5.04860090093668311608273372623, 5.32790836897096555983732668084, 5.96744368889847453387068579979, 6.76955885158912340897756950945, 6.87126538623379444026486653759, 7.30265771947486168528941346448, 8.052642541080546717924291648317, 8.528796002193558283113883461082
