L(s) = 1 | + 4-s + 2·7-s + 10·13-s + 16-s + 4·19-s − 10·25-s + 2·28-s + 10·31-s + 4·37-s − 2·43-s + 3·49-s + 10·52-s − 20·61-s + 64-s − 26·67-s + 4·73-s + 4·76-s − 20·79-s + 20·91-s + 16·97-s − 10·100-s − 26·103-s − 32·109-s + 2·112-s − 22·121-s + 10·124-s + 127-s + ⋯ |
L(s) = 1 | + 1/2·4-s + 0.755·7-s + 2.77·13-s + 1/4·16-s + 0.917·19-s − 2·25-s + 0.377·28-s + 1.79·31-s + 0.657·37-s − 0.304·43-s + 3/7·49-s + 1.38·52-s − 2.56·61-s + 1/8·64-s − 3.17·67-s + 0.468·73-s + 0.458·76-s − 2.25·79-s + 2.09·91-s + 1.62·97-s − 100-s − 2.56·103-s − 3.06·109-s + 0.188·112-s − 2·121-s + 0.898·124-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 142884 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 142884 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.350635247\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.350635247\) |
\(L(\frac{3}{2})\) |
|
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.358886306281919204370388916695, −8.682143010158560978147219777821, −8.360658565008429029479731162281, −7.77295226765590417072124632150, −7.66081859530557802153309309893, −6.72994634489814968752118634163, −6.29670817383847541122613475379, −5.74408802320171077841105153966, −5.61879984028395623126782300752, −4.35441080626390222166206464206, −4.30645127647783484429164317499, −3.34773629764867655679280376645, −2.89963569250000254763681159752, −1.68869994047015348511995782895, −1.25841763926025585348683837537,
1.25841763926025585348683837537, 1.68869994047015348511995782895, 2.89963569250000254763681159752, 3.34773629764867655679280376645, 4.30645127647783484429164317499, 4.35441080626390222166206464206, 5.61879984028395623126782300752, 5.74408802320171077841105153966, 6.29670817383847541122613475379, 6.72994634489814968752118634163, 7.66081859530557802153309309893, 7.77295226765590417072124632150, 8.360658565008429029479731162281, 8.682143010158560978147219777821, 9.358886306281919204370388916695