| L(s) = 1 | + 2-s − 3·3-s − 4-s + 4·5-s − 3·6-s + 2·7-s − 8-s + 6·9-s + 4·10-s − 4·11-s + 3·12-s + 8·13-s + 2·14-s − 12·15-s − 16-s + 2·17-s + 6·18-s + 2·19-s − 4·20-s − 6·21-s − 4·22-s − 10·23-s + 3·24-s + 3·25-s + 8·26-s − 10·27-s − 2·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.73·3-s − 1/2·4-s + 1.78·5-s − 1.22·6-s + 0.755·7-s − 0.353·8-s + 2·9-s + 1.26·10-s − 1.20·11-s + 0.866·12-s + 2.21·13-s + 0.534·14-s − 3.09·15-s − 1/4·16-s + 0.485·17-s + 1.41·18-s + 0.458·19-s − 0.894·20-s − 1.30·21-s − 0.852·22-s − 2.08·23-s + 0.612·24-s + 3/5·25-s + 1.56·26-s − 1.92·27-s − 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1860867 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1860867 ^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.166302621\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.166302621\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 + T )^{3} \) |
| 41 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 2 | $S_4\times C_2$ | \( 1 - T + p T^{2} - p T^{3} + p^{2} T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 - 4 T + 13 T^{2} - 36 T^{3} + 13 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - 2 T + p T^{2} + 4 T^{3} + p^{2} T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 4 T + 34 T^{2} + 84 T^{3} + 34 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 8 T + 53 T^{2} - 204 T^{3} + 53 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 2 T + 28 T^{2} - 6 T^{3} + 28 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 2 T + 51 T^{2} - 68 T^{3} + 51 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 10 T + 95 T^{2} + 476 T^{3} + 95 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 6 T + 60 T^{2} + 262 T^{3} + 60 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 2 T + 2 T^{2} - 132 T^{3} + 2 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 20 T + 228 T^{2} - 1646 T^{3} + 228 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 10 T + 10 T^{2} + 296 T^{3} + 10 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 4 T + 106 T^{2} - 384 T^{3} + 106 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 14 T + 3 p T^{2} - 1452 T^{3} + 3 p^{2} T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 8 T + 137 T^{2} + 976 T^{3} + 137 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 8 T + 188 T^{2} + 930 T^{3} + 188 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 12 T + 77 T^{2} - 632 T^{3} + 77 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 32 T + 550 T^{2} + 5712 T^{3} + 550 p T^{4} + 32 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 4 T + 120 T^{2} - 130 T^{3} + 120 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 20 T + 305 T^{2} + 3192 T^{3} + 305 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 14 T + 259 T^{2} + 2028 T^{3} + 259 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 14 T + 263 T^{2} - 2308 T^{3} + 263 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 12 T + 305 T^{2} + 2180 T^{3} + 305 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−11.90717999116278942174658298280, −11.76987345250675848387574567697, −11.30735358410147450614767662987, −11.18871132160682130416959767089, −10.59721445842523993615613944225, −10.35985445320676844721815151621, −10.25753485447878914241092098056, −9.567689349046625213550422192749, −9.363606218127519478073531872842, −9.113945133917980543859341876359, −8.235293305592256009725376556273, −8.123501962833172919429612627895, −7.50055697363095975182818161182, −7.30164885386734798260280588757, −6.34988165376075991129508494381, −6.02978154325852985056132767642, −5.79680387714073983696096153194, −5.67835400202771327628923698136, −5.40858957229402051237867835997, −4.56878405046191906398920494101, −4.12900087789777686630288841370, −4.10640427234977286403382053763, −2.85491513327109156080282679625, −1.99491455245009696356113689570, −1.28453518060447574247347756841,
1.28453518060447574247347756841, 1.99491455245009696356113689570, 2.85491513327109156080282679625, 4.10640427234977286403382053763, 4.12900087789777686630288841370, 4.56878405046191906398920494101, 5.40858957229402051237867835997, 5.67835400202771327628923698136, 5.79680387714073983696096153194, 6.02978154325852985056132767642, 6.34988165376075991129508494381, 7.30164885386734798260280588757, 7.50055697363095975182818161182, 8.123501962833172919429612627895, 8.235293305592256009725376556273, 9.113945133917980543859341876359, 9.363606218127519478073531872842, 9.567689349046625213550422192749, 10.25753485447878914241092098056, 10.35985445320676844721815151621, 10.59721445842523993615613944225, 11.18871132160682130416959767089, 11.30735358410147450614767662987, 11.76987345250675848387574567697, 11.90717999116278942174658298280
