| L(s) = 1 | − 3-s − 3·4-s + 9-s + 3·12-s + 4·13-s + 5·16-s − 8·19-s − 6·25-s − 27-s − 3·36-s + 12·37-s − 4·39-s − 8·43-s − 5·48-s − 12·52-s + 8·57-s + 4·61-s − 3·64-s + 8·67-s + 12·73-s + 6·75-s + 24·76-s − 32·79-s + 81-s − 36·97-s + 18·100-s − 16·103-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 3/2·4-s + 1/3·9-s + 0.866·12-s + 1.10·13-s + 5/4·16-s − 1.83·19-s − 6/5·25-s − 0.192·27-s − 1/2·36-s + 1.97·37-s − 0.640·39-s − 1.21·43-s − 0.721·48-s − 1.66·52-s + 1.05·57-s + 0.512·61-s − 3/8·64-s + 0.977·67-s + 1.40·73-s + 0.692·75-s + 2.75·76-s − 3.60·79-s + 1/9·81-s − 3.65·97-s + 9/5·100-s − 1.57·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64827 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64827 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( 1 + T \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 18 T + p T^{2} )^{2} \) |
| 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
−9.800766905340478341584155035885, −9.163045893798789171919999202805, −8.609463409035164758983775505912, −8.123916451859490481483222873179, −7.977686193886306753143677629185, −6.81422856027217448872168795499, −6.53484921474790582970192984243, −5.73267711633132091527857634664, −5.48510765590071063887274763802, −4.61914309368534933978095799917, −4.01671863219499060051377207353, −3.94328917656394473634657471263, −2.64839746138688614832708523229, −1.40786771277750203137263322761, 0,
1.40786771277750203137263322761, 2.64839746138688614832708523229, 3.94328917656394473634657471263, 4.01671863219499060051377207353, 4.61914309368534933978095799917, 5.48510765590071063887274763802, 5.73267711633132091527857634664, 6.53484921474790582970192984243, 6.81422856027217448872168795499, 7.977686193886306753143677629185, 8.123916451859490481483222873179, 8.609463409035164758983775505912, 9.163045893798789171919999202805, 9.800766905340478341584155035885
