| L(s) = 1 | − 3·5-s + 7-s + 4·13-s − 7·17-s − 3·23-s + 6·25-s − 5·29-s + 31-s − 3·35-s + 9·37-s − 3·41-s − 2·47-s − 8·49-s − 15·53-s − 3·59-s − 6·61-s − 12·65-s − 3·67-s − 5·71-s + 8·73-s + 8·79-s − 7·83-s + 21·85-s − 20·89-s + 4·91-s + 6·97-s − 5·101-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 0.377·7-s + 1.10·13-s − 1.69·17-s − 0.625·23-s + 6/5·25-s − 0.928·29-s + 0.179·31-s − 0.507·35-s + 1.47·37-s − 0.468·41-s − 0.291·47-s − 8/7·49-s − 2.06·53-s − 0.390·59-s − 0.768·61-s − 1.48·65-s − 0.366·67-s − 0.593·71-s + 0.936·73-s + 0.900·79-s − 0.768·83-s + 2.27·85-s − 2.11·89-s + 0.419·91-s + 0.609·97-s − 0.497·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{6} \cdot 5^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{6} \cdot 5^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{3} \) |
| 23 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 7 | $S_4\times C_2$ | \( 1 - T + 9 T^{2} + 2 T^{3} + 9 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) |
| 13 | $D_{6}$ | \( 1 - 4 T + 19 T^{2} - 88 T^{3} + 19 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 7 T + 55 T^{2} + 210 T^{3} + 55 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) |
| 29 | $S_4\times C_2$ | \( 1 + 5 T + 83 T^{2} + 286 T^{3} + 83 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - T + 61 T^{2} + 2 T^{3} + 61 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 9 T + 35 T^{2} + 10 T^{3} + 35 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 3 T + 83 T^{2} + 98 T^{3} + 83 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 65 T^{2} - 64 T^{3} + 65 p T^{4} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 2 T + 93 T^{2} + 60 T^{3} + 93 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 15 T + 191 T^{2} + 1394 T^{3} + 191 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 3 T + 137 T^{2} + 418 T^{3} + 137 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 6 T + 131 T^{2} + 548 T^{3} + 131 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 3 T + 101 T^{2} - 94 T^{3} + 101 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 5 T + 97 T^{2} + 822 T^{3} + 97 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 8 T + 191 T^{2} - 960 T^{3} + 191 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 8 T + 173 T^{2} - 816 T^{3} + 173 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 7 T + 141 T^{2} + 570 T^{3} + 141 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 20 T + 271 T^{2} + 2568 T^{3} + 271 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 6 T + 239 T^{2} - 980 T^{3} + 239 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.44191747325154427460449529997, −6.83698205477255831995354859298, −6.77394541119418648931024100013, −6.67055817457487175700325222750, −6.24301268106250960007944455334, −6.12946035307044600550146753907, −6.09075752819745182226568489550, −5.56244863025258672545486286450, −5.26616227098596369125426967902, −5.21456347473478356879408605719, −4.70296791044613724770648936635, −4.49967364234310744151434439764, −4.48169048533474181930145425617, −4.24118635825929282192183911847, −3.83862411669441500919544368829, −3.68043383904575163085924917775, −3.41953208281192358433935289372, −3.13657711012142433437236660438, −2.94333940834748171982124458032, −2.41603491968295324538270967156, −2.25021745626474247248974625197, −2.03314993585933821874340093308, −1.35616069398772325168687171697, −1.28993354376377019864292309931, −1.07730084674812779794514208972, 0, 0, 0,
1.07730084674812779794514208972, 1.28993354376377019864292309931, 1.35616069398772325168687171697, 2.03314993585933821874340093308, 2.25021745626474247248974625197, 2.41603491968295324538270967156, 2.94333940834748171982124458032, 3.13657711012142433437236660438, 3.41953208281192358433935289372, 3.68043383904575163085924917775, 3.83862411669441500919544368829, 4.24118635825929282192183911847, 4.48169048533474181930145425617, 4.49967364234310744151434439764, 4.70296791044613724770648936635, 5.21456347473478356879408605719, 5.26616227098596369125426967902, 5.56244863025258672545486286450, 6.09075752819745182226568489550, 6.12946035307044600550146753907, 6.24301268106250960007944455334, 6.67055817457487175700325222750, 6.77394541119418648931024100013, 6.83698205477255831995354859298, 7.44191747325154427460449529997
