| L(s) = 1 | + 3·3-s − 3·5-s − 7-s + 6·9-s − 11-s + 3·13-s − 9·15-s − 17-s − 6·19-s − 3·21-s + 7·23-s + 6·25-s + 10·27-s + 18·29-s − 6·31-s − 3·33-s + 3·35-s + 13·37-s + 9·39-s + 41-s − 18·45-s + 18·47-s − 4·49-s − 3·51-s + 11·53-s + 3·55-s − 18·57-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 1.34·5-s − 0.377·7-s + 2·9-s − 0.301·11-s + 0.832·13-s − 2.32·15-s − 0.242·17-s − 1.37·19-s − 0.654·21-s + 1.45·23-s + 6/5·25-s + 1.92·27-s + 3.34·29-s − 1.07·31-s − 0.522·33-s + 0.507·35-s + 2.13·37-s + 1.44·39-s + 0.156·41-s − 2.68·45-s + 2.62·47-s − 4/7·49-s − 0.420·51-s + 1.51·53-s + 0.404·55-s − 2.38·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 5^{3} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 5^{3} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.381895061\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.381895061\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{3} \) |
| 5 | $C_1$ | \( ( 1 + T )^{3} \) |
| 13 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 7 | $S_4\times C_2$ | \( 1 + T + 5 T^{2} + 30 T^{3} + 5 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + T + 17 T^{2} + 38 T^{3} + 17 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + T + 19 T^{2} - 42 T^{3} + 19 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 6 T + 41 T^{2} + 164 T^{3} + 41 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 7 T + 53 T^{2} - 194 T^{3} + 53 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{3} \) |
| 31 | $S_4\times C_2$ | \( 1 + 6 T + 77 T^{2} + 340 T^{3} + 77 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 13 T + 3 p T^{2} - 646 T^{3} + 3 p^{2} T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - T + 91 T^{2} - 6 T^{3} + 91 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 17 T^{2} + 128 T^{3} + 17 p T^{4} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 18 T + 221 T^{2} - 1756 T^{3} + 221 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 11 T + 167 T^{2} - 1162 T^{3} + 167 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 8 T + 129 T^{2} - 816 T^{3} + 129 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 9 T + 71 T^{2} - 254 T^{3} + 71 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 4 T + 137 T^{2} + 408 T^{3} + 137 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 11 T + 237 T^{2} - 1530 T^{3} + 237 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 6 T + 119 T^{2} + 532 T^{3} + 119 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 5 T + 189 T^{2} + 854 T^{3} + 189 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 8 T + 201 T^{2} + 1200 T^{3} + 201 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 11 T + 275 T^{2} - 1954 T^{3} + 275 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 25 T + 467 T^{2} + 5094 T^{3} + 467 p T^{4} + 25 p^{2} T^{5} + p^{3} T^{6} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.971830511855868230321472726576, −7.36073982642915528446343846116, −7.20364564924307615954881828914, −7.16095273234630354814137336318, −6.71861675375331202160585115946, −6.57300077115648274002320638326, −6.45904383810418388995304530904, −5.77360541882474378872534421570, −5.72614577425642521402039754486, −5.48678608698776094214658094556, −4.73951412884620596790996057238, −4.66824120053552995048248855056, −4.56264670257622861828310876341, −4.03755433158935442183035864471, −3.95292608918586095060011219546, −3.86922971270337248606651619421, −3.15655109938814187858902939817, −3.09695922553871635091731817801, −2.86065649458417664913200029732, −2.55918212345189131286272712587, −2.10432438310711905633720823861, −2.00122415000973466499252920835, −1.04127366519358831036233467674, −0.947868163224890282926069706082, −0.59468445627953642683902386555,
0.59468445627953642683902386555, 0.947868163224890282926069706082, 1.04127366519358831036233467674, 2.00122415000973466499252920835, 2.10432438310711905633720823861, 2.55918212345189131286272712587, 2.86065649458417664913200029732, 3.09695922553871635091731817801, 3.15655109938814187858902939817, 3.86922971270337248606651619421, 3.95292608918586095060011219546, 4.03755433158935442183035864471, 4.56264670257622861828310876341, 4.66824120053552995048248855056, 4.73951412884620596790996057238, 5.48678608698776094214658094556, 5.72614577425642521402039754486, 5.77360541882474378872534421570, 6.45904383810418388995304530904, 6.57300077115648274002320638326, 6.71861675375331202160585115946, 7.16095273234630354814137336318, 7.20364564924307615954881828914, 7.36073982642915528446343846116, 7.971830511855868230321472726576
