| L(s) = 1 | + 3·2-s + 6·4-s − 2·5-s + 10·8-s − 6·10-s + 5·11-s − 5·13-s + 15·16-s + 7·17-s − 12·19-s − 12·20-s + 15·22-s + 7·23-s − 10·25-s − 15·26-s + 22·29-s − 3·31-s + 21·32-s + 21·34-s + 3·37-s − 36·38-s − 20·40-s + 7·41-s + 5·43-s + 30·44-s + 21·46-s + 11·47-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 3·4-s − 0.894·5-s + 3.53·8-s − 1.89·10-s + 1.50·11-s − 1.38·13-s + 15/4·16-s + 1.69·17-s − 2.75·19-s − 2.68·20-s + 3.19·22-s + 1.45·23-s − 2·25-s − 2.94·26-s + 4.08·29-s − 0.538·31-s + 3.71·32-s + 3.60·34-s + 0.493·37-s − 5.83·38-s − 3.16·40-s + 1.09·41-s + 0.762·43-s + 4.52·44-s + 3.09·46-s + 1.60·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{6} \cdot 7^{9}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{6} \cdot 7^{9}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(26.33387846\) |
| \(L(\frac12)\) |
\(\approx\) |
\(26.33387846\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $A_4\times C_2$ | \( 1 + 2 T + 14 T^{2} + 19 T^{3} + 14 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 - 5 T + p T^{2} - 13 T^{3} + p^{2} T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $A_4\times C_2$ | \( 1 + 5 T + 17 T^{2} + 33 T^{3} + 17 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 - 7 T + 37 T^{2} - 147 T^{3} + 37 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $A_4\times C_2$ | \( 1 + 12 T + 98 T^{2} + 499 T^{3} + 98 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 - 7 T + 83 T^{2} - 329 T^{3} + 83 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 - 22 T + 8 p T^{2} - 1527 T^{3} + 8 p^{2} T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 + 3 T + 47 T^{2} + 47 T^{3} + 47 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 - 3 T + 65 T^{2} - 223 T^{3} + 65 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 - 7 T + 81 T^{2} - 483 T^{3} + 81 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 - 5 T + 121 T^{2} - 389 T^{3} + 121 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 - 11 T + 95 T^{2} - 487 T^{3} + 95 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 - 2 T + 116 T^{2} - 85 T^{3} + 116 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 - 26 T + 386 T^{2} - 3601 T^{3} + 386 p T^{4} - 26 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 - 5 T + 35 T^{2} + 201 T^{3} + 35 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 - 26 T + 410 T^{2} - 4017 T^{3} + 410 p T^{4} - 26 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 - 4 T + 146 T^{2} - 539 T^{3} + 146 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 + T + 63 T^{2} + 649 T^{3} + 63 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 + 3 T + 65 T^{2} - 575 T^{3} + 65 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 - 13 T + 275 T^{2} - 2159 T^{3} + 275 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 + 18 T + 284 T^{2} + 2587 T^{3} + 284 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 + 10 T + 112 T^{2} + 79 T^{3} + 112 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.05969204235580089796237945679, −6.70189794531770153337652559737, −6.68630631065975217673349909790, −6.46716022913654258924601360124, −6.09683226127236084144855959750, −5.88679714662035213996999866928, −5.61216932545719023492761208222, −5.36487885030797892058785977371, −5.22496820526841505796365333136, −4.86830295601892324414513011087, −4.57327468704983209536527526755, −4.31078402922224877298442085201, −4.13490077767670549750352056511, −3.94005539330466432097213482452, −3.88980590607911664533344188541, −3.60907157660242365362797656726, −2.90509893316901272222651451144, −2.88784158704806723109977877064, −2.77798989812842336589987259633, −2.21828783848300003613317813348, −2.03875126568331283517614646068, −1.86560812791810320835059610785, −1.04144660641383905019223582640, −0.74232776409697478440538777523, −0.71322398598342770072200662208,
0.71322398598342770072200662208, 0.74232776409697478440538777523, 1.04144660641383905019223582640, 1.86560812791810320835059610785, 2.03875126568331283517614646068, 2.21828783848300003613317813348, 2.77798989812842336589987259633, 2.88784158704806723109977877064, 2.90509893316901272222651451144, 3.60907157660242365362797656726, 3.88980590607911664533344188541, 3.94005539330466432097213482452, 4.13490077767670549750352056511, 4.31078402922224877298442085201, 4.57327468704983209536527526755, 4.86830295601892324414513011087, 5.22496820526841505796365333136, 5.36487885030797892058785977371, 5.61216932545719023492761208222, 5.88679714662035213996999866928, 6.09683226127236084144855959750, 6.46716022913654258924601360124, 6.68630631065975217673349909790, 6.70189794531770153337652559737, 7.05969204235580089796237945679
