| L(s) = 1 | − 3·2-s + 6·4-s − 10·8-s + 15·16-s − 3·19-s + 27-s − 21·32-s + 3·37-s + 9·38-s − 3·47-s − 3·54-s + 28·64-s − 9·74-s − 18·76-s + 9·94-s + 6·108-s + 3·121-s + 127-s − 36·128-s + 131-s + 137-s + 139-s + 18·148-s + 149-s + 151-s + 30·152-s + 157-s + ⋯ |
| L(s) = 1 | − 3·2-s + 6·4-s − 10·8-s + 15·16-s − 3·19-s + 27-s − 21·32-s + 3·37-s + 9·38-s − 3·47-s − 3·54-s + 28·64-s − 9·74-s − 18·76-s + 9·94-s + 6·108-s + 3·121-s + 127-s − 36·128-s + 131-s + 137-s + 139-s + 18·148-s + 149-s + 151-s + 30·152-s + 157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3182338573\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3182338573\) |
| \(L(1)\) |
|
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 3 | $C_6$ | \( 1 - T^{3} + T^{6} \) |
| 7 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 13 | $C_6$ | \( 1 - T^{3} + T^{6} \) |
| 17 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 23 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 29 | $C_6$ | \( 1 - T^{3} + T^{6} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )^{3} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 47 | $C_2$ | \( ( 1 + T + T^{2} )^{3} \) |
| 53 | $C_6$ | \( 1 - T^{3} + T^{6} \) |
| 59 | $C_6$ | \( 1 - T^{3} + T^{6} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 67 | $C_6$ | \( 1 - T^{3} + T^{6} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 73 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.920433166717209397823752313459, −7.74696446579968495356410257820, −7.22259940738282807331510472649, −6.94738330976620738177121407657, −6.93300801448237260386046237234, −6.57910072727121896937510160842, −6.32276338722519711534820452520, −6.23220699731047492022741219470, −5.89163009425620121811726394799, −5.87858761736417789078448667577, −5.29325736723530133678251333539, −4.98360230009489519366366666265, −4.61408884470067770113623007625, −4.33720108578422471704685902859, −3.91515814680727749871202078643, −3.74698871582306586024589522387, −3.01541659037606063468058618145, −2.97525884109578343826211530157, −2.84304511510068920946380533510, −2.29358438037025803274176720683, −2.03319230706837636392119405231, −1.73590910442619845156342712456, −1.55059409854462001210166804268, −0.790026702321987899437065594714, −0.53475195151724166465600046100,
0.53475195151724166465600046100, 0.790026702321987899437065594714, 1.55059409854462001210166804268, 1.73590910442619845156342712456, 2.03319230706837636392119405231, 2.29358438037025803274176720683, 2.84304511510068920946380533510, 2.97525884109578343826211530157, 3.01541659037606063468058618145, 3.74698871582306586024589522387, 3.91515814680727749871202078643, 4.33720108578422471704685902859, 4.61408884470067770113623007625, 4.98360230009489519366366666265, 5.29325736723530133678251333539, 5.87858761736417789078448667577, 5.89163009425620121811726394799, 6.23220699731047492022741219470, 6.32276338722519711534820452520, 6.57910072727121896937510160842, 6.93300801448237260386046237234, 6.94738330976620738177121407657, 7.22259940738282807331510472649, 7.74696446579968495356410257820, 7.920433166717209397823752313459
