| L(s) = 1 | + 3-s + 4-s + 12-s + 13-s + 3·19-s − 2·25-s − 27-s + 3·37-s + 39-s + 43-s + 52-s + 3·57-s − 61-s − 64-s − 2·75-s + 3·76-s − 4·79-s − 81-s − 3·97-s − 2·100-s − 2·103-s − 108-s + 3·111-s + 121-s + 127-s + 129-s + 131-s + ⋯ |
| L(s) = 1 | + 3-s + 4-s + 12-s + 13-s + 3·19-s − 2·25-s − 27-s + 3·37-s + 39-s + 43-s + 52-s + 3·57-s − 61-s − 64-s − 2·75-s + 3·76-s − 4·79-s − 81-s − 3·97-s − 2·100-s − 2·103-s − 108-s + 3·111-s + 121-s + 127-s + 129-s + 131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3651921 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3651921 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.382493726\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.382493726\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | | \( 1 \) |
| 13 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 79 | $C_1$ | \( ( 1 + T )^{4} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 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
−9.539617976675782593102825891352, −9.185815733142517086475517805462, −8.958738313129502722774150561012, −8.247329423283318454193391845671, −7.87941653306926686946060124983, −7.82124782762048437819464464021, −7.21220611272151546723678212208, −7.15948050781961457142094209113, −6.34966520918478172905305014541, −6.00644609599014517815311886888, −5.56447710362294550137150329293, −5.53308812304842322226745067563, −4.50397730533051305641566800075, −4.14855489657331897356883493198, −3.68644264518180289501742837283, −3.13732270115115354470991765878, −2.63614960444786369680539498936, −2.59085406668557991351579058436, −1.53942532651469422276912997261, −1.27181276358664632801289333205,
1.27181276358664632801289333205, 1.53942532651469422276912997261, 2.59085406668557991351579058436, 2.63614960444786369680539498936, 3.13732270115115354470991765878, 3.68644264518180289501742837283, 4.14855489657331897356883493198, 4.50397730533051305641566800075, 5.53308812304842322226745067563, 5.56447710362294550137150329293, 6.00644609599014517815311886888, 6.34966520918478172905305014541, 7.15948050781961457142094209113, 7.21220611272151546723678212208, 7.82124782762048437819464464021, 7.87941653306926686946060124983, 8.247329423283318454193391845671, 8.958738313129502722774150561012, 9.185815733142517086475517805462, 9.539617976675782593102825891352
