| L(s) = 1 | + 4·7-s − 16-s − 4·31-s + 6·49-s + 4·79-s + 4·97-s − 4·112-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s − 16·217-s + ⋯ |
| L(s) = 1 | + 4·7-s − 16-s − 4·31-s + 6·49-s + 4·79-s + 4·97-s − 4·112-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s − 16·217-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.092336267\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.092336267\) |
| \(L(1)\) |
|
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 | $C_2^2$ | \( 1 + T^{4} \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 11 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
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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
−6.72715719898056972431047264189, −6.66295405610610940077042814572, −6.31533481245987775461089904822, −6.07737540406506055558584639307, −5.69870925495695370086835407673, −5.55883319347678730542009997196, −5.50878672213563103491044415998, −5.11069575605308624946143039482, −4.94437388266083209238158860047, −4.88193245207303930816239053249, −4.70749714626511209537759667430, −4.54929799002007938829681284848, −4.07929457943493870407039745471, −4.05726779644681044988906897009, −3.73399661856818561901693396929, −3.38879317227613203082748904615, −3.31805948659218553400063318145, −2.92998011807298066742639966130, −2.25826607525284778639689372752, −2.19604389965158858972674501915, −2.02049087937541360844441506960, −1.88093645591966394024054027820, −1.52703396807628706455572392143, −1.30176333535926952156344348775, −0.74133253750177856940002867049,
0.74133253750177856940002867049, 1.30176333535926952156344348775, 1.52703396807628706455572392143, 1.88093645591966394024054027820, 2.02049087937541360844441506960, 2.19604389965158858972674501915, 2.25826607525284778639689372752, 2.92998011807298066742639966130, 3.31805948659218553400063318145, 3.38879317227613203082748904615, 3.73399661856818561901693396929, 4.05726779644681044988906897009, 4.07929457943493870407039745471, 4.54929799002007938829681284848, 4.70749714626511209537759667430, 4.88193245207303930816239053249, 4.94437388266083209238158860047, 5.11069575605308624946143039482, 5.50878672213563103491044415998, 5.55883319347678730542009997196, 5.69870925495695370086835407673, 6.07737540406506055558584639307, 6.31533481245987775461089904822, 6.66295405610610940077042814572, 6.72715719898056972431047264189
