| L(s) = 1 | + 2·3-s + 4·7-s − 9-s − 2·11-s − 8·13-s + 4·17-s − 2·19-s + 8·21-s + 8·23-s − 4·27-s + 6·29-s − 8·31-s − 4·33-s − 8·37-s − 16·39-s + 2·41-s + 14·43-s + 4·47-s − 49-s + 8·51-s − 16·53-s − 4·57-s + 2·59-s + 10·61-s − 4·63-s + 10·67-s + 16·69-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1.51·7-s − 1/3·9-s − 0.603·11-s − 2.21·13-s + 0.970·17-s − 0.458·19-s + 1.74·21-s + 1.66·23-s − 0.769·27-s + 1.11·29-s − 1.43·31-s − 0.696·33-s − 1.31·37-s − 2.56·39-s + 0.312·41-s + 2.13·43-s + 0.583·47-s − 1/7·49-s + 1.12·51-s − 2.19·53-s − 0.529·57-s + 0.260·59-s + 1.28·61-s − 0.503·63-s + 1.22·67-s + 1.92·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{21} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{21} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.702137962\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.702137962\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $S_4\times C_2$ | \( 1 - 2 T + 5 T^{2} - 8 T^{3} + 5 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - 4 T + 17 T^{2} - 36 T^{3} + 17 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 2 T + 13 T^{2} + 36 T^{3} + 13 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 8 T + 47 T^{2} + 192 T^{3} + 47 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 4 T + 35 T^{2} - 104 T^{3} + 35 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 2 T + 21 T^{2} - 28 T^{3} + 21 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 8 T + 81 T^{2} - 372 T^{3} + 81 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{3} \) |
| 31 | $S_4\times C_2$ | \( 1 + 8 T + 61 T^{2} + 368 T^{3} + 61 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 8 T + 79 T^{2} + 320 T^{3} + 79 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 2 T + 71 T^{2} + 20 T^{3} + 71 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 14 T + 149 T^{2} - 1104 T^{3} + 149 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 4 T + 113 T^{2} - 260 T^{3} + 113 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 16 T + 231 T^{2} + 1776 T^{3} + 231 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 2 T + 141 T^{2} - 132 T^{3} + 141 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 10 T + 155 T^{2} - 1212 T^{3} + 155 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 10 T + 141 T^{2} - 736 T^{3} + 141 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 12 T + 197 T^{2} - 1384 T^{3} + 197 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 20 T + 3 p T^{2} - 1736 T^{3} + 3 p^{2} T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 16 T + 269 T^{2} + 2400 T^{3} + 269 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 30 T + 509 T^{2} - 5504 T^{3} + 509 p T^{4} - 30 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 10 T + 151 T^{2} + 684 T^{3} + 151 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 28 T + 531 T^{2} - 6040 T^{3} + 531 p T^{4} - 28 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.81060347054039361457807521024, −7.42144560526949132968217235751, −7.34468239360412700955953681541, −7.14499451226538099767245060507, −6.71824801536930619791864554633, −6.39221396156625667082469358163, −6.37899238525800611842886862586, −5.61830266651023568008051107114, −5.48299834837077359173317608808, −5.37546130847900761710707410237, −4.96295336133700988609108189185, −4.96214297633742132834470643333, −4.54998967188077940003743899656, −4.42443223908866239701528360667, −3.89922648554738463165874093992, −3.52231814485166403415978200085, −3.27487240026946847647069572218, −3.03592046566992109231946390370, −2.75454919647689904465728617948, −2.22212677528352204485524357865, −2.17470275109637980377164936634, −2.02364358752162649883944355094, −1.37865335869888555938128545745, −0.78401633119309830528696581442, −0.50548820658250741172095060195,
0.50548820658250741172095060195, 0.78401633119309830528696581442, 1.37865335869888555938128545745, 2.02364358752162649883944355094, 2.17470275109637980377164936634, 2.22212677528352204485524357865, 2.75454919647689904465728617948, 3.03592046566992109231946390370, 3.27487240026946847647069572218, 3.52231814485166403415978200085, 3.89922648554738463165874093992, 4.42443223908866239701528360667, 4.54998967188077940003743899656, 4.96214297633742132834470643333, 4.96295336133700988609108189185, 5.37546130847900761710707410237, 5.48299834837077359173317608808, 5.61830266651023568008051107114, 6.37899238525800611842886862586, 6.39221396156625667082469358163, 6.71824801536930619791864554633, 7.14499451226538099767245060507, 7.34468239360412700955953681541, 7.42144560526949132968217235751, 7.81060347054039361457807521024
