| L(s) = 1 | + 3·2-s + 3·3-s + 6·4-s − 3·5-s + 9·6-s + 6·7-s + 10·8-s + 6·9-s − 9·10-s + 18·12-s + 18·14-s − 9·15-s + 15·16-s − 7·17-s + 18·18-s − 19-s − 18·20-s + 18·21-s + 3·23-s + 30·24-s + 6·25-s + 10·27-s + 36·28-s − 27·30-s + 16·31-s + 21·32-s − 21·34-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 1.73·3-s + 3·4-s − 1.34·5-s + 3.67·6-s + 2.26·7-s + 3.53·8-s + 2·9-s − 2.84·10-s + 5.19·12-s + 4.81·14-s − 2.32·15-s + 15/4·16-s − 1.69·17-s + 4.24·18-s − 0.229·19-s − 4.02·20-s + 3.92·21-s + 0.625·23-s + 6.12·24-s + 6/5·25-s + 1.92·27-s + 6.80·28-s − 4.92·30-s + 2.87·31-s + 3.71·32-s − 3.60·34-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
| \(L(1)\) |
\(\approx\) |
\(68.86291070\) |
| \(L(\frac12)\) |
\(\approx\) |
\(68.86291070\) |
| \(L(\frac{3}{2})\) |
|
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 | $A_4\times C_2$ | \( 1 - 6 T + 26 T^{2} - 71 T^{3} + 26 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 + 26 T^{2} + 7 T^{3} + 26 p T^{4} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 + 7 T + 3 p T^{2} + 231 T^{3} + 3 p^{2} T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $A_4\times C_2$ | \( 1 + T + 41 T^{2} + 51 T^{3} + 41 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 - 3 T + 65 T^{2} - 139 T^{3} + 65 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 - 4 T^{2} - 7 p T^{3} - 4 p T^{4} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 - 16 T + 162 T^{2} - 1075 T^{3} + 162 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 - 20 T + 200 T^{2} - 1383 T^{3} + 200 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 - 17 T + 203 T^{2} - 1507 T^{3} + 203 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 - 9 T + 107 T^{2} - 605 T^{3} + 107 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 - 7 T + 92 T^{2} - 707 T^{3} + 92 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 - 3 T + 141 T^{2} - 305 T^{3} + 141 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 - 7 T + 163 T^{2} - 819 T^{3} + 163 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 - 5 T + 189 T^{2} - 611 T^{3} + 189 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 - 17 T + 281 T^{2} - 2349 T^{3} + 281 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 + 7 T + 3 p T^{2} + 945 T^{3} + 3 p^{2} T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 - 3 T + 89 T^{2} - 5 T^{3} + 89 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 - 17 T + 247 T^{2} - 2099 T^{3} + 247 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 - 6 T + 128 T^{2} - 647 T^{3} + 128 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 + 3 T + 123 T^{2} - 249 T^{3} + 123 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 - 24 T + 434 T^{2} - 4825 T^{3} + 434 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.54318631181925867240888082226, −6.91828353460135478628495402233, −6.84439824012692477389042504955, −6.61893601820834221094173120816, −6.24886470852570385520367636333, −5.99916422967516959369248192442, −5.96828891131067998743911088657, −5.15176396055753321546959436186, −5.13077868039064078358683882620, −5.04304401098518053194329168294, −4.49901256574526282666274096927, −4.44602588341079490136554012209, −4.32024046025299993374760019575, −3.99766922502014349560981145629, −3.97251512219645676968407269603, −3.68222504970372363600392112628, −3.03814864175516232075485417109, −2.88220139527315090851176651076, −2.63766328641122596027233868394, −2.33423224006432297351616982705, −2.17938605003118437250197858413, −2.07214080818837533350997329564, −1.09726644194426819937869221615, −1.00203575540890503852424381996, −0.930336160147899085661994311602,
0.930336160147899085661994311602, 1.00203575540890503852424381996, 1.09726644194426819937869221615, 2.07214080818837533350997329564, 2.17938605003118437250197858413, 2.33423224006432297351616982705, 2.63766328641122596027233868394, 2.88220139527315090851176651076, 3.03814864175516232075485417109, 3.68222504970372363600392112628, 3.97251512219645676968407269603, 3.99766922502014349560981145629, 4.32024046025299993374760019575, 4.44602588341079490136554012209, 4.49901256574526282666274096927, 5.04304401098518053194329168294, 5.13077868039064078358683882620, 5.15176396055753321546959436186, 5.96828891131067998743911088657, 5.99916422967516959369248192442, 6.24886470852570385520367636333, 6.61893601820834221094173120816, 6.84439824012692477389042504955, 6.91828353460135478628495402233, 7.54318631181925867240888082226
