L(s) = 1 | − 3·13-s + 3·19-s + 27-s + 3·43-s + 6·61-s + 3·67-s − 3·73-s − 6·79-s − 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
L(s) = 1 | − 3·13-s + 3·19-s + 27-s + 3·43-s + 6·61-s + 3·67-s − 3·73-s − 6·79-s − 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7566594888\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7566594888\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T^{3} + T^{6} \) |
| 19 | \( ( 1 - T + T^{2} )^{3} \) |
good | 5 | \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \) |
| 7 | \( ( 1 - T^{3} + T^{6} )^{2} \) |
| 11 | \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \) |
| 13 | \( ( 1 + T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \) |
| 17 | \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \) |
| 23 | \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \) |
| 29 | \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \) |
| 31 | \( ( 1 - T^{3} + T^{6} )^{2} \) |
| 37 | \( ( 1 + T^{3} + T^{6} )^{2} \) |
| 41 | \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \) |
| 43 | \( ( 1 - T + T^{2} )^{3}( 1 - T^{3} + T^{6} ) \) |
| 47 | \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \) |
| 53 | \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \) |
| 59 | \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \) |
| 61 | \( ( 1 - T )^{6}( 1 + T^{3} + T^{6} ) \) |
| 67 | \( ( 1 - T + T^{2} )^{3}( 1 - T^{3} + T^{6} ) \) |
| 71 | \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \) |
| 73 | \( ( 1 + T + T^{2} )^{3}( 1 + T^{3} + T^{6} ) \) |
| 79 | \( ( 1 + T )^{6}( 1 - T^{3} + T^{6} ) \) |
| 83 | \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \) |
| 89 | \( ( 1 - T^{3} + T^{6} )( 1 + T^{3} + T^{6} ) \) |
| 97 | \( ( 1 + T^{3} + T^{6} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.58203363676118635319975095362, −5.25422472975716030756694238752, −5.19957207990768060614414757973, −5.19202263432930426657095381939, −5.06941569722518177602294777764, −4.90754189264064532091556541060, −4.51270146605892299673393448812, −4.50620694711157630334599757815, −4.21334276750321749810035647422, −4.15852599832802457999956165105, −4.00247537330916088204788532662, −3.69656193780226036249956006023, −3.52335292919550285782092668387, −3.45249173387382402618476299658, −2.91483501260857518765720066120, −2.87578115028088518878629118637, −2.85137561760291189083279524415, −2.59443019736734478909813973360, −2.46609609339575195967818244721, −2.25367904216824911079160065794, −1.95280307646931823337428521543, −1.73102939741099338568308066525, −1.22796604635395947468663186980, −0.969150725369289885523021710750, −0.897879797936330743191016736577,
0.897879797936330743191016736577, 0.969150725369289885523021710750, 1.22796604635395947468663186980, 1.73102939741099338568308066525, 1.95280307646931823337428521543, 2.25367904216824911079160065794, 2.46609609339575195967818244721, 2.59443019736734478909813973360, 2.85137561760291189083279524415, 2.87578115028088518878629118637, 2.91483501260857518765720066120, 3.45249173387382402618476299658, 3.52335292919550285782092668387, 3.69656193780226036249956006023, 4.00247537330916088204788532662, 4.15852599832802457999956165105, 4.21334276750321749810035647422, 4.50620694711157630334599757815, 4.51270146605892299673393448812, 4.90754189264064532091556541060, 5.06941569722518177602294777764, 5.19202263432930426657095381939, 5.19957207990768060614414757973, 5.25422472975716030756694238752, 5.58203363676118635319975095362
Plot not available for L-functions of degree greater than 10.