| L(s) = 1 | + 9·3-s + 11·5-s + 54·9-s − 35·11-s + 62·13-s + 99·15-s + 48·17-s + 202·19-s − 216·23-s − 62·25-s + 270·27-s + 53·29-s + 95·31-s − 315·33-s + 262·37-s + 558·39-s + 244·41-s − 360·43-s + 594·45-s + 210·47-s + 432·51-s + 393·53-s − 385·55-s + 1.81e3·57-s − 1.14e3·59-s − 70·61-s + 682·65-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.983·5-s + 2·9-s − 0.959·11-s + 1.32·13-s + 1.70·15-s + 0.684·17-s + 2.43·19-s − 1.95·23-s − 0.495·25-s + 1.92·27-s + 0.339·29-s + 0.550·31-s − 1.66·33-s + 1.16·37-s + 2.29·39-s + 0.929·41-s − 1.27·43-s + 1.96·45-s + 0.651·47-s + 1.18·51-s + 1.01·53-s − 0.943·55-s + 4.22·57-s − 2.52·59-s − 0.146·61-s + 1.30·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(21.88495295\) |
| \(L(\frac12)\) |
\(\approx\) |
\(21.88495295\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 - p T )^{3} \) |
| 7 | | \( 1 \) |
| good | 5 | $S_4\times C_2$ | \( 1 - 11 T + 183 T^{2} - 1514 T^{3} + 183 p^{3} T^{4} - 11 p^{6} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 35 T + 2625 T^{2} + 102734 T^{3} + 2625 p^{3} T^{4} + 35 p^{6} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 62 T + 7016 T^{2} - 253976 T^{3} + 7016 p^{3} T^{4} - 62 p^{6} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 48 T + 12339 T^{2} - 358752 T^{3} + 12339 p^{3} T^{4} - 48 p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 202 T + 32858 T^{2} - 3004840 T^{3} + 32858 p^{3} T^{4} - 202 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 216 T + 35829 T^{2} + 3675600 T^{3} + 35829 p^{3} T^{4} + 216 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 53 T + 52695 T^{2} - 3410210 T^{3} + 52695 p^{3} T^{4} - 53 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 95 T + 79372 T^{2} - 5648467 T^{3} + 79372 p^{3} T^{4} - 95 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 262 T + 166048 T^{2} - 26591324 T^{3} + 166048 p^{3} T^{4} - 262 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 244 T + 187983 T^{2} - 33933832 T^{3} + 187983 p^{3} T^{4} - 244 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 360 T + 166158 T^{2} + 38975294 T^{3} + 166158 p^{3} T^{4} + 360 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 210 T + 64953 T^{2} - 48724788 T^{3} + 64953 p^{3} T^{4} - 210 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 393 T + 365895 T^{2} - 83847930 T^{3} + 365895 p^{3} T^{4} - 393 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 1143 T + 12699 p T^{2} + 369027450 T^{3} + 12699 p^{4} T^{4} + 1143 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 70 T + 340043 T^{2} - 52853660 T^{3} + 340043 p^{3} T^{4} + 70 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 628 T + 597326 T^{2} - 6055990 p T^{3} + 597326 p^{3} T^{4} - 628 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 318 T + 742929 T^{2} + 256167372 T^{3} + 742929 p^{3} T^{4} + 318 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 988 T + 1162696 T^{2} - 625490474 T^{3} + 1162696 p^{3} T^{4} - 988 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 861 T + 1221216 T^{2} + 655056821 T^{3} + 1221216 p^{3} T^{4} + 861 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 519 T + 1583745 T^{2} + 545598870 T^{3} + 1583745 p^{3} T^{4} + 519 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 1766 T + 2392827 T^{2} - 2476945964 T^{3} + 2392827 p^{3} T^{4} - 1766 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 19 T + 2168419 T^{2} + 10094878 T^{3} + 2168419 p^{3} T^{4} - 19 p^{6} T^{5} + p^{9} T^{6} \) |
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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
−7.69303512813081196216099799400, −7.41992028302480036749019169888, −7.37616857535695739731354432730, −6.97612083470199563298865907453, −6.37570283519257414353681887382, −6.30320346918028220602283856005, −6.14231053055773284935287291528, −5.67258523172997287200256595533, −5.54547036376210995587926514553, −5.41003023002926887305493104610, −4.76098081866589802598218651933, −4.56442995314931697813478200805, −4.47718289851516277056247245263, −3.72123875561222702666223629492, −3.61825740202547922492559899269, −3.61333664914902805674795878980, −2.91869573765371475351539456218, −2.90005117890120903478917245731, −2.64741704376159549904435017706, −1.93278482860642723353166887668, −1.90147094551224845955926339891, −1.74249318066661409041432220034, −0.963459406538530712534210879917, −0.912029186720007407272693273454, −0.46412701501602308708775725105,
0.46412701501602308708775725105, 0.912029186720007407272693273454, 0.963459406538530712534210879917, 1.74249318066661409041432220034, 1.90147094551224845955926339891, 1.93278482860642723353166887668, 2.64741704376159549904435017706, 2.90005117890120903478917245731, 2.91869573765371475351539456218, 3.61333664914902805674795878980, 3.61825740202547922492559899269, 3.72123875561222702666223629492, 4.47718289851516277056247245263, 4.56442995314931697813478200805, 4.76098081866589802598218651933, 5.41003023002926887305493104610, 5.54547036376210995587926514553, 5.67258523172997287200256595533, 6.14231053055773284935287291528, 6.30320346918028220602283856005, 6.37570283519257414353681887382, 6.97612083470199563298865907453, 7.37616857535695739731354432730, 7.41992028302480036749019169888, 7.69303512813081196216099799400
