| L(s) = 1 | − 2·7-s − 4·13-s − 4·19-s − 2·25-s − 2·31-s − 2·37-s + 49-s + 4·61-s − 4·67-s + 4·73-s − 2·79-s + 8·91-s + 2·97-s + 127-s + 131-s + 8·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + 4·175-s + 179-s + ⋯ |
| L(s) = 1 | − 2·7-s − 4·13-s − 4·19-s − 2·25-s − 2·31-s − 2·37-s + 49-s + 4·61-s − 4·67-s + 4·73-s − 2·79-s + 8·91-s + 2·97-s + 127-s + 131-s + 8·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + 4·175-s + 179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1534962864\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1534962864\) |
| \(L(1)\) |
|
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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 13 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 5 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 19 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 23 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 29 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 53 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 67 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
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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
−6.27430715572912427650405739212, −6.24416042010170190429775089463, −5.89611231572809821723577915260, −5.58289663049439040748794080625, −5.55823483494291729604294943642, −5.52889268128206788673382843406, −4.95367153981536741996987440815, −4.88303045572056338877109824624, −4.67995890143454758617231203456, −4.52974414712229999087273067037, −4.37704213761387596433419758463, −3.86845787164824539350683676732, −3.77693984117800121878460254179, −3.62691245289476280781552849135, −3.60081084122729698048832291354, −3.15770158382926835527668024724, −2.76762151087367438784065601839, −2.53221226693260905516649512459, −2.40587755118734405216640018201, −2.29548991181695169886372243160, −1.95788007539599408314378141195, −1.80005152585176942636838475972, −1.55570479553752003792279177240, −0.37053477952606348518084171674, −0.32913077067220079392047046110,
0.32913077067220079392047046110, 0.37053477952606348518084171674, 1.55570479553752003792279177240, 1.80005152585176942636838475972, 1.95788007539599408314378141195, 2.29548991181695169886372243160, 2.40587755118734405216640018201, 2.53221226693260905516649512459, 2.76762151087367438784065601839, 3.15770158382926835527668024724, 3.60081084122729698048832291354, 3.62691245289476280781552849135, 3.77693984117800121878460254179, 3.86845787164824539350683676732, 4.37704213761387596433419758463, 4.52974414712229999087273067037, 4.67995890143454758617231203456, 4.88303045572056338877109824624, 4.95367153981536741996987440815, 5.52889268128206788673382843406, 5.55823483494291729604294943642, 5.58289663049439040748794080625, 5.89611231572809821723577915260, 6.24416042010170190429775089463, 6.27430715572912427650405739212
