L(s) = 1 | − 32·25-s + 20·49-s − 40·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 32·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + ⋯ |
L(s) = 1 | − 6.39·25-s + 20/7·49-s − 3.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 2.46·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + 0.0646·239-s + 0.0644·241-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3211327097\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3211327097\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
good | 5 | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{4} \) |
| 11 | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 - p T^{2} )^{8} \) |
| 29 | \( ( 1 - p T^{2} )^{8} \) |
| 31 | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 - 10 T + p T^{2} )^{4}( 1 + 10 T + p T^{2} )^{4} \) |
| 41 | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 + 74 T^{2} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 - 68 T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 + 104 T^{2} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 - 106 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 - 68 T^{2} + p^{2} T^{4} )^{4} \) |
| 89 | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 - 170 T^{2} + p^{2} T^{4} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.56498202363410212107600969051, −3.34430355010386565986538541540, −3.30951505729650825148447340301, −3.22639201520828456143403045164, −2.98295262326766249944305955094, −2.91205459944702433095297288580, −2.76610625983105064212462993682, −2.58576901583342993414847325921, −2.40581754919085392437872356724, −2.39705750281945139763259569666, −2.34134910228465659913863843605, −2.21142405776857027013473220163, −1.98307531043502486974613685021, −1.87428668967976392852758189548, −1.77977472075599113627954142308, −1.77393592739132982523766256422, −1.64925633260620435728135738152, −1.37485016163143301901602881663, −1.20376026882145597606574023517, −1.12603938735552139829015927113, −0.75514871342674771733566827741, −0.71987696797490580908168032134, −0.47541861450766493886441763221, −0.35494873524933593420990089930, −0.05000685128759226873769255427,
0.05000685128759226873769255427, 0.35494873524933593420990089930, 0.47541861450766493886441763221, 0.71987696797490580908168032134, 0.75514871342674771733566827741, 1.12603938735552139829015927113, 1.20376026882145597606574023517, 1.37485016163143301901602881663, 1.64925633260620435728135738152, 1.77393592739132982523766256422, 1.77977472075599113627954142308, 1.87428668967976392852758189548, 1.98307531043502486974613685021, 2.21142405776857027013473220163, 2.34134910228465659913863843605, 2.39705750281945139763259569666, 2.40581754919085392437872356724, 2.58576901583342993414847325921, 2.76610625983105064212462993682, 2.91205459944702433095297288580, 2.98295262326766249944305955094, 3.22639201520828456143403045164, 3.30951505729650825148447340301, 3.34430355010386565986538541540, 3.56498202363410212107600969051
Plot not available for L-functions of degree greater than 10.