L(s) = 1 | + 2·25-s + 2·37-s + 2·43-s − 2·67-s − 2·79-s + 2·109-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯ |
L(s) = 1 | + 2·25-s + 2·37-s + 2·43-s − 2·67-s − 2·79-s + 2·109-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.250796336\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.250796336\) |
\(L(1)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
good | 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 11 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 67 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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
−9.546241743492637231301743564817, −9.257921809794012541669276835668, −8.865857949801841562228746163395, −8.694103321015126369281764338354, −7.990575680104474115606634308158, −7.79251351648400939042685222173, −7.22425857804538120847758920325, −7.08962852102827203835123985182, −6.34061604149370236202950670022, −6.19813868057511161059301623153, −5.67759528043369136618398569354, −5.24536026167022527772398936435, −4.61772661579888641664611156362, −4.43807658020438802710731254167, −3.91280339917285512320307417121, −3.26571334090267834472061865824, −2.63217218719953035275204927225, −2.57510274776060613681995637119, −1.50487015914138607533452316670, −0.942561592259345294468296927486,
0.942561592259345294468296927486, 1.50487015914138607533452316670, 2.57510274776060613681995637119, 2.63217218719953035275204927225, 3.26571334090267834472061865824, 3.91280339917285512320307417121, 4.43807658020438802710731254167, 4.61772661579888641664611156362, 5.24536026167022527772398936435, 5.67759528043369136618398569354, 6.19813868057511161059301623153, 6.34061604149370236202950670022, 7.08962852102827203835123985182, 7.22425857804538120847758920325, 7.79251351648400939042685222173, 7.990575680104474115606634308158, 8.694103321015126369281764338354, 8.865857949801841562228746163395, 9.257921809794012541669276835668, 9.546241743492637231301743564817