| L(s) = 1 | − 8·13-s − 20·25-s − 2·49-s + 56·61-s − 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 12·169-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 + ⋯ |
| L(s) = 1 | − 2.21·13-s − 4·25-s − 2/7·49-s + 7.17·61-s − 4·121-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 − 0.923·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.242516485\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.242516485\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 122 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \) |
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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
−5.81170964435799979733276339506, −5.61955653724020712870269442155, −5.55993180108530986227102079730, −5.51101063437025849128201526937, −5.21432803452624977476462048041, −5.01860074071379287355490215441, −4.82772161255169869662093473819, −4.54209176860720060275450311583, −4.37518461489084211009165768490, −4.00330661170997031533123748273, −3.94155058520768099546722139192, −3.86532793729983517217203733019, −3.51296938052169774684316213301, −3.51153567569268180409215748451, −2.92824938117631338596895936260, −2.77856712573096466156397874208, −2.47730890764769768314090476273, −2.39459460167001029275778154554, −2.12642650724445793238324389250, −1.86284509353554694976982083407, −1.79486262186254845523773444626, −1.35350069480209126611802736575, −0.78485583488529101672547895178, −0.59157377942880437839567845842, −0.20611579941423764907065112433,
0.20611579941423764907065112433, 0.59157377942880437839567845842, 0.78485583488529101672547895178, 1.35350069480209126611802736575, 1.79486262186254845523773444626, 1.86284509353554694976982083407, 2.12642650724445793238324389250, 2.39459460167001029275778154554, 2.47730890764769768314090476273, 2.77856712573096466156397874208, 2.92824938117631338596895936260, 3.51153567569268180409215748451, 3.51296938052169774684316213301, 3.86532793729983517217203733019, 3.94155058520768099546722139192, 4.00330661170997031533123748273, 4.37518461489084211009165768490, 4.54209176860720060275450311583, 4.82772161255169869662093473819, 5.01860074071379287355490215441, 5.21432803452624977476462048041, 5.51101063437025849128201526937, 5.55993180108530986227102079730, 5.61955653724020712870269442155, 5.81170964435799979733276339506
