| L(s) = 1 | − 3·2-s + 3·3-s + 6·4-s − 3·5-s − 9·6-s + 10·7-s − 10·8-s + 6·9-s + 9·10-s + 8·11-s + 18·12-s − 30·14-s − 9·15-s + 15·16-s − 3·17-s − 18·18-s + 11·19-s − 18·20-s + 30·21-s − 24·22-s − 9·23-s − 30·24-s + 6·25-s + 10·27-s + 60·28-s + 27·30-s + 16·31-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 1.73·3-s + 3·4-s − 1.34·5-s − 3.67·6-s + 3.77·7-s − 3.53·8-s + 2·9-s + 2.84·10-s + 2.41·11-s + 5.19·12-s − 8.01·14-s − 2.32·15-s + 15/4·16-s − 0.727·17-s − 4.24·18-s + 2.52·19-s − 4.02·20-s + 6.54·21-s − 5.11·22-s − 1.87·23-s − 6.12·24-s + 6/5·25-s + 1.92·27-s + 11.3·28-s + 4.92·30-s + 2.87·31-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
| \(L(1)\) |
\(\approx\) |
\(9.266346922\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.266346922\) |
| \(L(\frac{3}{2})\) |
|
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 | $A_4\times C_2$ | \( 1 - 10 T + 52 T^{2} - 169 T^{3} + 52 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 - 8 T + 52 T^{2} - 189 T^{3} + 52 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 + 3 T + 5 T^{2} - 37 T^{3} + 5 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $A_4\times C_2$ | \( 1 - 11 T + 53 T^{2} - 207 T^{3} + 53 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 + 9 T + 89 T^{2} + 427 T^{3} + 89 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 + 24 T^{2} + 189 T^{3} + 24 p T^{4} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 - 16 T + 176 T^{2} - 1131 T^{3} + 176 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 - 10 T + 100 T^{2} - 517 T^{3} + 100 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 - 5 T + 87 T^{2} - 243 T^{3} + 87 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 + 3 T + 69 T^{2} + 385 T^{3} + 69 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 - 3 T + 32 T^{2} + 277 T^{3} + 32 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 + 17 T + 239 T^{2} + 1873 T^{3} + 239 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 - 11 T + 61 T^{2} - 219 T^{3} + 61 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 - 11 T + 193 T^{2} - 1313 T^{3} + 193 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 - 15 T + 213 T^{2} - 2009 T^{3} + 213 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $C_6$ | \( 1 - 13 T + 183 T^{2} - 1833 T^{3} + 183 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 + T - 21 T^{2} + 523 T^{3} - 21 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 + 3 T + 219 T^{2} + 447 T^{3} + 219 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 - 12 T + 276 T^{2} - 1965 T^{3} + 276 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 - T + 27 T^{2} + 831 T^{3} + 27 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 + 6 T + 2 T^{2} - 1397 T^{3} + 2 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.71662317176023335906064658460, −7.26041152112841639190394269275, −6.92902606917092097273464364928, −6.90736348814798056870876331202, −6.40776626153954386104477563313, −6.38540316801648913873259455169, −6.10926138365097511262815063521, −5.38851262820302847999246857815, −5.23594998332920912722346105013, −5.14432823229774307438050967178, −4.46177676844643666981726300711, −4.45761261521035848546731509054, −4.37761402083730720886328663899, −3.81135494100607934206314630857, −3.66095749334016510783173793465, −3.64236301453157518844017280125, −2.80356100072758271647331762737, −2.69638848853330192864087700278, −2.53418287910930099257937495119, −1.84662072632633505149244198466, −1.84579632741894035076039638235, −1.55986512162802956482381395759, −1.07412132227436685376947686481, −0.912579351530342337874080420322, −0.798634481318780423679832679399,
0.798634481318780423679832679399, 0.912579351530342337874080420322, 1.07412132227436685376947686481, 1.55986512162802956482381395759, 1.84579632741894035076039638235, 1.84662072632633505149244198466, 2.53418287910930099257937495119, 2.69638848853330192864087700278, 2.80356100072758271647331762737, 3.64236301453157518844017280125, 3.66095749334016510783173793465, 3.81135494100607934206314630857, 4.37761402083730720886328663899, 4.45761261521035848546731509054, 4.46177676844643666981726300711, 5.14432823229774307438050967178, 5.23594998332920912722346105013, 5.38851262820302847999246857815, 6.10926138365097511262815063521, 6.38540316801648913873259455169, 6.40776626153954386104477563313, 6.90736348814798056870876331202, 6.92902606917092097273464364928, 7.26041152112841639190394269275, 7.71662317176023335906064658460
