| L(s) = 1 | − 3·2-s − 3·3-s + 6·4-s + 3·5-s + 9·6-s + 8·7-s − 10·8-s + 6·9-s − 9·10-s + 2·11-s − 18·12-s − 24·14-s − 9·15-s + 15·16-s + 3·17-s − 18·18-s − 3·19-s + 18·20-s − 24·21-s − 6·22-s + 11·23-s + 30·24-s + 6·25-s − 10·27-s + 48·28-s + 2·29-s + 27·30-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 1.73·3-s + 3·4-s + 1.34·5-s + 3.67·6-s + 3.02·7-s − 3.53·8-s + 2·9-s − 2.84·10-s + 0.603·11-s − 5.19·12-s − 6.41·14-s − 2.32·15-s + 15/4·16-s + 0.727·17-s − 4.24·18-s − 0.688·19-s + 4.02·20-s − 5.23·21-s − 1.27·22-s + 2.29·23-s + 6.12·24-s + 6/5·25-s − 1.92·27-s + 9.07·28-s + 0.371·29-s + 4.92·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 5^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 5^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.174102586\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.174102586\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| 5 | $C_1$ | \( ( 1 - T )^{3} \) |
| 13 | | \( 1 \) |
| good | 7 | $A_4\times C_2$ | \( 1 - 8 T + 40 T^{2} - 125 T^{3} + 40 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 - 2 T + 32 T^{2} - 43 T^{3} + 32 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 - 3 T + 5 T^{2} - 5 T^{3} + 5 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $A_4\times C_2$ | \( 1 + 3 T + 39 T^{2} + 87 T^{3} + 39 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 - 11 T + 79 T^{2} - 393 T^{3} + 79 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 - 2 T + 44 T^{2} + 11 T^{3} + 44 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 + 86 T^{2} - 7 T^{3} + 86 p T^{4} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 + 8 T + 18 T^{2} - 151 T^{3} + 18 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 - 13 T + 163 T^{2} - 1095 T^{3} + 163 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 + 5 T + 121 T^{2} + 431 T^{3} + 121 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 - 7 T - 20 T^{2} + 525 T^{3} - 20 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 - 19 T + 207 T^{2} - 1665 T^{3} + 207 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 - 23 T + 323 T^{2} - 2881 T^{3} + 323 p T^{4} - 23 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 + 9 T + 161 T^{2} + 1069 T^{3} + 161 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 - 13 T + 213 T^{2} - 1519 T^{3} + 213 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 - 3 T + 209 T^{2} - 413 T^{3} + 209 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 - 17 T + 271 T^{2} - 2525 T^{3} + 271 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 + 25 T + 387 T^{2} + 4075 T^{3} + 387 p T^{4} + 25 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 - 20 T + 268 T^{2} - 2943 T^{3} + 268 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 - 21 T + 239 T^{2} - 1981 T^{3} + 239 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 - 6 T + 240 T^{2} - 857 T^{3} + 240 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.39435251571733651167964716335, −7.09504654172793873878816244274, −6.82166034402038784341262868804, −6.69419979985273590605932872251, −6.29582689258910157218267742706, −6.25221180093788425512539656800, −5.85337484722427987022647775922, −5.47966051912567994504883850190, −5.37834810517168410600239602866, −5.29081711185365225161889679601, −4.79931493252929415400373701477, −4.77747145864549295479881626556, −4.67282761443468632564715306582, −3.79017163757963711311596710788, −3.73125113921214843641479629738, −3.72181912380671196159341034477, −2.59426246880366383518821996085, −2.57440981653289131985948537652, −2.47041256071497167141201274386, −1.82479984396048362744413212778, −1.59279655512156638292963990338, −1.56738241539015144873789532707, −0.927535903782753781866149163498, −0.871005823326946464219427606020, −0.65148629031827451791568146291,
0.65148629031827451791568146291, 0.871005823326946464219427606020, 0.927535903782753781866149163498, 1.56738241539015144873789532707, 1.59279655512156638292963990338, 1.82479984396048362744413212778, 2.47041256071497167141201274386, 2.57440981653289131985948537652, 2.59426246880366383518821996085, 3.72181912380671196159341034477, 3.73125113921214843641479629738, 3.79017163757963711311596710788, 4.67282761443468632564715306582, 4.77747145864549295479881626556, 4.79931493252929415400373701477, 5.29081711185365225161889679601, 5.37834810517168410600239602866, 5.47966051912567994504883850190, 5.85337484722427987022647775922, 6.25221180093788425512539656800, 6.29582689258910157218267742706, 6.69419979985273590605932872251, 6.82166034402038784341262868804, 7.09504654172793873878816244274, 7.39435251571733651167964716335
