| L(s) = 1 | + 3·3-s − 8-s + 6·9-s − 3·24-s + 10·27-s − 3·29-s + 3·41-s − 6·72-s + 15·81-s − 9·87-s + 3·103-s + 9·123-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | + 3·3-s − 8-s + 6·9-s − 3·24-s + 10·27-s − 3·29-s + 3·41-s − 6·72-s + 15·81-s − 9·87-s + 3·103-s + 9·123-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 29^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 29^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(4.273807836\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.273807836\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - T )^{3} \) |
| 5 | | \( 1 \) |
| 29 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 2 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 7 | $C_6$ | \( 1 - T^{3} + T^{6} \) |
| 11 | $C_6$ | \( 1 - T^{3} + T^{6} \) |
| 13 | $C_6$ | \( 1 - T^{3} + T^{6} \) |
| 17 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )^{3} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 47 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 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_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 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 | $C_6$ | \( 1 - T^{3} + T^{6} \) |
| 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
−8.283628838011331338515175036407, −7.86323795441089878409619645631, −7.86218075976287764752333334966, −7.44643075740033179845349751065, −7.31224669850524849697821884818, −7.30625998703915144637154656499, −6.82883412006812277965661625110, −6.43713359714796277460327852542, −6.17209459992901541821773673578, −5.88609013077102886402845440676, −5.71246823902183627437253769463, −5.07190150257792623463188221684, −4.93368819348352006813103738464, −4.52470512231292599012275165847, −4.23692971573057199033517471435, −3.80980449569617653538362104569, −3.72530078415204495070506315368, −3.51742713403905625313206686108, −3.17069456319245308203767496122, −2.67132526055191886201146298813, −2.55739365404473325363480450571, −2.23321786640478887660271663919, −1.94074726955905458397913084652, −1.42222125349456273924850061714, −1.02541387754140829046272838321,
1.02541387754140829046272838321, 1.42222125349456273924850061714, 1.94074726955905458397913084652, 2.23321786640478887660271663919, 2.55739365404473325363480450571, 2.67132526055191886201146298813, 3.17069456319245308203767496122, 3.51742713403905625313206686108, 3.72530078415204495070506315368, 3.80980449569617653538362104569, 4.23692971573057199033517471435, 4.52470512231292599012275165847, 4.93368819348352006813103738464, 5.07190150257792623463188221684, 5.71246823902183627437253769463, 5.88609013077102886402845440676, 6.17209459992901541821773673578, 6.43713359714796277460327852542, 6.82883412006812277965661625110, 7.30625998703915144637154656499, 7.31224669850524849697821884818, 7.44643075740033179845349751065, 7.86218075976287764752333334966, 7.86323795441089878409619645631, 8.283628838011331338515175036407
