| L(s) = 1 | − 2·11-s − 8·13-s + 12·23-s − 3·25-s + 40·37-s + 20·47-s + 49-s + 8·59-s + 32·71-s − 12·73-s − 18·83-s − 14·97-s + 68·107-s + 8·109-s + 23·121-s + 127-s + 131-s + 137-s + 139-s + 16·143-s + 149-s + 151-s + 157-s + 163-s + 167-s + 42·169-s + 173-s + ⋯ |
| L(s) = 1 | − 0.603·11-s − 2.21·13-s + 2.50·23-s − 3/5·25-s + 6.57·37-s + 2.91·47-s + 1/7·49-s + 1.04·59-s + 3.79·71-s − 1.40·73-s − 1.97·83-s − 1.42·97-s + 6.57·107-s + 0.766·109-s + 2.09·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.33·143-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3.23·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.217104047\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.217104047\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^3$ | \( 1 + 3 T^{2} - 16 T^{4} + 3 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $C_2^3$ | \( 1 - T^{2} - 48 T^{4} - p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 + T - 10 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 4 T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^3$ | \( 1 - 6 T^{2} - 805 T^{4} - 6 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_2^3$ | \( 1 - 49 T^{2} + 1440 T^{4} - 49 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{4} \) |
| 41 | $C_2^3$ | \( 1 - 30 T^{2} - 781 T^{4} - 30 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $C_2^3$ | \( 1 - 34 T^{2} - 693 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 - 10 T + 53 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 93 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 - 82 T^{2} + 2235 T^{4} - 82 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 18 T + 187 T^{2} - 18 p T^{3} + p^{2} T^{4} )( 1 + 18 T + 187 T^{2} + 18 p T^{3} + p^{2} T^{4} ) \) |
| 83 | $C_2^2$ | \( ( 1 + 9 T - 2 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 126 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 7 T - 48 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.24529168814103502829748233192, −6.09915705861933772953636236383, −5.75737327333538372591264318711, −5.65030947506385510011315830108, −5.57979359843194714747805097218, −5.34121432691070845918174292140, −4.92620690427994051918874938635, −4.91826298786824211979870499475, −4.48904074832209216485787042667, −4.44863135133404490789666772261, −4.33183883921067075881604374366, −4.08495196972266975303787195601, −3.94846073397057183246952910627, −3.17796644283662085581490496269, −3.17681865538028980820722463136, −3.12371203391964638530795861330, −2.82811973858215806239206656138, −2.34252387284092740259132653578, −2.26878126027490913674364691779, −2.24978732367439604043979062366, −1.98737457987623921168783652039, −1.05405903061460403587325698342, −1.01041727365449291567429926171, −0.74984779901455562888970875210, −0.47050646583233183624933664408,
0.47050646583233183624933664408, 0.74984779901455562888970875210, 1.01041727365449291567429926171, 1.05405903061460403587325698342, 1.98737457987623921168783652039, 2.24978732367439604043979062366, 2.26878126027490913674364691779, 2.34252387284092740259132653578, 2.82811973858215806239206656138, 3.12371203391964638530795861330, 3.17681865538028980820722463136, 3.17796644283662085581490496269, 3.94846073397057183246952910627, 4.08495196972266975303787195601, 4.33183883921067075881604374366, 4.44863135133404490789666772261, 4.48904074832209216485787042667, 4.91826298786824211979870499475, 4.92620690427994051918874938635, 5.34121432691070845918174292140, 5.57979359843194714747805097218, 5.65030947506385510011315830108, 5.75737327333538372591264318711, 6.09915705861933772953636236383, 6.24529168814103502829748233192
