| L(s) = 1 | + 2·4-s + 8·19-s + 10·25-s + 20·43-s − 14·49-s − 8·64-s − 28·67-s + 8·73-s + 16·76-s + 20·97-s + 20·100-s − 14·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 26·169-s + 40·172-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | + 4-s + 1.83·19-s + 2·25-s + 3.04·43-s − 2·49-s − 64-s − 3.42·67-s + 0.936·73-s + 1.83·76-s + 2.03·97-s + 2·100-s − 1.27·121-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 − 2·169-s + 3.04·172-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.976191812\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.976191812\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \) |
| 13 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 41 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \) |
| 43 | $C_2^2$ | \( ( 1 - 10 T + 57 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 59 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 6 T - 23 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \) |
| 61 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 14 T + 129 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 18 T + 241 T^{2} - 18 p T^{3} + p^{2} T^{4} )( 1 + 18 T + 241 T^{2} + 18 p T^{3} + p^{2} T^{4} ) \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2}( 1 + 18 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 10 T + 3 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.53021583784779964880884880040, −7.44547345673844390365579788830, −7.11666027261264462856554877039, −6.94610833045269602799251803276, −6.67421754769981061490747647340, −6.41393758956509253734278504980, −6.31847905302640599106477930363, −5.89701918867828366602938440360, −5.62950742355998522429812203905, −5.60893794420405578656809842938, −5.27082070028871773508167765680, −4.78967384552911708289730842874, −4.64679650058616018927627704801, −4.50783540317841986114746882219, −4.26228403567267934635315587836, −3.66233335730430501892377238223, −3.38007721658756800872919200214, −3.27485815069797705908059316950, −2.84349512476922018580285030845, −2.70939775202098723860756402694, −2.34556713359145431020197491604, −1.91944243790322267206906112850, −1.43649121692833088804410698127, −1.15949047844928478056530366707, −0.59878789389094428760888348338,
0.59878789389094428760888348338, 1.15949047844928478056530366707, 1.43649121692833088804410698127, 1.91944243790322267206906112850, 2.34556713359145431020197491604, 2.70939775202098723860756402694, 2.84349512476922018580285030845, 3.27485815069797705908059316950, 3.38007721658756800872919200214, 3.66233335730430501892377238223, 4.26228403567267934635315587836, 4.50783540317841986114746882219, 4.64679650058616018927627704801, 4.78967384552911708289730842874, 5.27082070028871773508167765680, 5.60893794420405578656809842938, 5.62950742355998522429812203905, 5.89701918867828366602938440360, 6.31847905302640599106477930363, 6.41393758956509253734278504980, 6.67421754769981061490747647340, 6.94610833045269602799251803276, 7.11666027261264462856554877039, 7.44547345673844390365579788830, 7.53021583784779964880884880040
