| L(s) = 1 | + 3·2-s + 6·4-s + 10·8-s + 15·16-s − 3·29-s + 21·32-s − 9·58-s − 3·59-s + 28·64-s − 3·79-s − 3·103-s − 18·116-s − 9·118-s − 125-s + 127-s + 36·128-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 9·158-s + 163-s + 167-s + 3·169-s + 173-s + ⋯ |
| L(s) = 1 | + 3·2-s + 6·4-s + 10·8-s + 15·16-s − 3·29-s + 21·32-s − 9·58-s − 3·59-s + 28·64-s − 3·79-s − 3·103-s − 18·116-s − 9·118-s − 125-s + 127-s + 36·128-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 9·158-s + 163-s + 167-s + 3·169-s + 173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{15}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{15}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(10.69165515\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.69165515\) |
| \(L(1)\) |
|
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 | $C_1$ | \( ( 1 - T )^{3} \) |
| 3 | | \( 1 \) |
| good | 5 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 7 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 11 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 29 | $C_2$ | \( ( 1 + T + T^{2} )^{3} \) |
| 31 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 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_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 53 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 59 | $C_2$ | \( ( 1 + T + T^{2} )^{3} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 73 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 79 | $C_2$ | \( ( 1 + T + T^{2} )^{3} \) |
| 83 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 97 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
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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
−8.248657873933707644746410608189, −7.73345599899026477150537858469, −7.72393546548683963720999475523, −7.40854626637518321855706664911, −7.12055370706312490695017985342, −6.94248096036538192995115210229, −6.65725733622306575615451844640, −6.32108185533757391508376788048, −6.06061239198158276948949017509, −5.69889848288708428412709476834, −5.66636221727447880132136545191, −5.42263622262079504387285885681, −5.07756413982387066463753188454, −4.66013257151085150313069505521, −4.49927007277098114818278023877, −4.24325911359177373557490627449, −3.83876074959046488307903148485, −3.66227289075139448412540855009, −3.38931478857536976794364975057, −2.91860718261155005754866185041, −2.74296885371371209127082111201, −2.41861866997818736355041107047, −1.78267083195715027470346593981, −1.66252062183210165214534472484, −1.33245676145799885624549174458,
1.33245676145799885624549174458, 1.66252062183210165214534472484, 1.78267083195715027470346593981, 2.41861866997818736355041107047, 2.74296885371371209127082111201, 2.91860718261155005754866185041, 3.38931478857536976794364975057, 3.66227289075139448412540855009, 3.83876074959046488307903148485, 4.24325911359177373557490627449, 4.49927007277098114818278023877, 4.66013257151085150313069505521, 5.07756413982387066463753188454, 5.42263622262079504387285885681, 5.66636221727447880132136545191, 5.69889848288708428412709476834, 6.06061239198158276948949017509, 6.32108185533757391508376788048, 6.65725733622306575615451844640, 6.94248096036538192995115210229, 7.12055370706312490695017985342, 7.40854626637518321855706664911, 7.72393546548683963720999475523, 7.73345599899026477150537858469, 8.248657873933707644746410608189
