| L(s) = 1 | − 2·5-s + 2·7-s − 10·11-s − 4·13-s + 8·17-s + 3·19-s − 4·23-s − 4·25-s + 6·29-s + 18·31-s − 4·35-s + 12·37-s + 14·41-s + 10·43-s − 8·47-s + 6·49-s + 10·53-s + 20·55-s + 8·59-s + 8·65-s + 16·73-s − 20·77-s + 22·79-s − 18·83-s − 16·85-s + 6·89-s − 8·91-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.755·7-s − 3.01·11-s − 1.10·13-s + 1.94·17-s + 0.688·19-s − 0.834·23-s − 4/5·25-s + 1.11·29-s + 3.23·31-s − 0.676·35-s + 1.97·37-s + 2.18·41-s + 1.52·43-s − 1.16·47-s + 6/7·49-s + 1.37·53-s + 2.69·55-s + 1.04·59-s + 0.992·65-s + 1.87·73-s − 2.27·77-s + 2.47·79-s − 1.97·83-s − 1.73·85-s + 0.635·89-s − 0.838·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.506517971\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.506517971\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 5 | $S_4\times C_2$ | \( 1 + 2 T + 8 T^{2} + 22 T^{3} + 8 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - 2 T - 2 T^{2} + 4 p T^{3} - 2 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 10 T + 58 T^{2} + 222 T^{3} + 58 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 4 T - 5 T^{2} - 56 T^{3} - 5 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 8 T + 36 T^{2} - 120 T^{3} + 36 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 4 T + 41 T^{2} + 200 T^{3} + 41 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 6 T + 59 T^{2} - 244 T^{3} + 59 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{3} \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{3} \) |
| 41 | $S_4\times C_2$ | \( 1 - 14 T + 139 T^{2} - 1116 T^{3} + 139 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 10 T + 138 T^{2} - 856 T^{3} + 138 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 8 T + 102 T^{2} + 432 T^{3} + 102 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 10 T + 67 T^{2} - 396 T^{3} + 67 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 8 T + T^{2} + 336 T^{3} + p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 116 T^{2} - 190 T^{3} + 116 p T^{4} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 41 T^{2} - 256 T^{3} + 41 p T^{4} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 53 T^{2} + 256 T^{3} + 53 p T^{4} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 16 T + 224 T^{2} - 2214 T^{3} + 224 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 22 T + 349 T^{2} - 3412 T^{3} + 349 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 18 T + 317 T^{2} + 2932 T^{3} + 317 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 6 T + 239 T^{2} - 1028 T^{3} + 239 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 22 T + 319 T^{2} - 3220 T^{3} + 319 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.80216187180713824272078678956, −7.61943206764324692940821498562, −7.50803358849870518383037130127, −7.43981927524138002216424981819, −6.84365789378159225727909678767, −6.44635628110503708359749117748, −6.35165498121577252597861329262, −5.84287751626937806874084055094, −5.63866573038222832661748356988, −5.45592564043770390255880050309, −5.28361910548045623745760877140, −4.81988911569034139102130886755, −4.73173280935912438099276100532, −4.27174954043337402186507663683, −4.24131640754213853412986322404, −3.90028888095897396760909962970, −3.22189354944770394763854600378, −3.03066104773356851706765226911, −2.86735129804313463807383461774, −2.49590539940599076898388139371, −2.11307061209173312415203079123, −2.09888302162659560389018976184, −0.946421504137900704437002137957, −0.70946273429085315645708596144, −0.66389323873369476547373894364,
0.66389323873369476547373894364, 0.70946273429085315645708596144, 0.946421504137900704437002137957, 2.09888302162659560389018976184, 2.11307061209173312415203079123, 2.49590539940599076898388139371, 2.86735129804313463807383461774, 3.03066104773356851706765226911, 3.22189354944770394763854600378, 3.90028888095897396760909962970, 4.24131640754213853412986322404, 4.27174954043337402186507663683, 4.73173280935912438099276100532, 4.81988911569034139102130886755, 5.28361910548045623745760877140, 5.45592564043770390255880050309, 5.63866573038222832661748356988, 5.84287751626937806874084055094, 6.35165498121577252597861329262, 6.44635628110503708359749117748, 6.84365789378159225727909678767, 7.43981927524138002216424981819, 7.50803358849870518383037130127, 7.61943206764324692940821498562, 7.80216187180713824272078678956
