| L(s) = 1 | − 3·3-s − 3·5-s + 6·9-s − 4·11-s + 6·13-s + 9·15-s + 2·17-s + 3·19-s − 4·23-s + 6·25-s − 10·27-s + 2·29-s − 8·31-s + 12·33-s + 6·37-s − 18·39-s − 2·41-s − 12·43-s − 18·45-s + 4·47-s − 7·49-s − 6·51-s + 2·53-s + 12·55-s − 9·57-s − 12·59-s + 14·61-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 1.34·5-s + 2·9-s − 1.20·11-s + 1.66·13-s + 2.32·15-s + 0.485·17-s + 0.688·19-s − 0.834·23-s + 6/5·25-s − 1.92·27-s + 0.371·29-s − 1.43·31-s + 2.08·33-s + 0.986·37-s − 2.88·39-s − 0.312·41-s − 1.82·43-s − 2.68·45-s + 0.583·47-s − 49-s − 0.840·51-s + 0.274·53-s + 1.61·55-s − 1.19·57-s − 1.56·59-s + 1.79·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 5^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 5^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.440756965\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.440756965\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| 19 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 7 | $S_4\times C_2$ | \( 1 + p T^{2} - 16 T^{3} + p^{2} T^{4} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 4 T + 7 T^{2} - 8 T^{3} + 7 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $D_{6}$ | \( 1 - 6 T + 37 T^{2} - 120 T^{3} + 37 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 2 T + 27 T^{2} - 52 T^{3} + 27 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 4 T + 49 T^{2} + 152 T^{3} + 49 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 2 T + 41 T^{2} + 40 T^{3} + 41 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 8 T + 89 T^{2} + 448 T^{3} + 89 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 6 T + 109 T^{2} - 408 T^{3} + 109 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 2 T + 93 T^{2} + 200 T^{3} + 93 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 12 T + 163 T^{2} + 1024 T^{3} + 163 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 4 T + 121 T^{2} - 344 T^{3} + 121 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 2 T + 135 T^{2} - 196 T^{3} + 135 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 12 T + 169 T^{2} + 1128 T^{3} + 169 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 14 T + 151 T^{2} - 1132 T^{3} + 151 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 4 T + 105 T^{2} + 408 T^{3} + 105 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 8 T + 109 T^{2} + 368 T^{3} + 109 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{3} \) |
| 79 | $S_4\times C_2$ | \( 1 - 4 T + 141 T^{2} - 504 T^{3} + 141 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 12 T - 35 T^{2} - 1464 T^{3} - 35 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 2 T + 141 T^{2} - 152 T^{3} + 141 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 30 T + 577 T^{2} - 6664 T^{3} + 577 p T^{4} - 30 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.47641027159507669913966469524, −7.14895817382775261480952973757, −6.76375131074506409994416460273, −6.69937079791830339775835528432, −6.24567174371663540764457455098, −6.16938542931036424871859811240, −5.95015214540643568874222890407, −5.56715452265414334711730264005, −5.45471184241227539161713615691, −5.12633724426551357687722013577, −4.90857428746292522172906084883, −4.60893112676358788196324467499, −4.53121116695540434977518117393, −3.99384385244426311562481846001, −3.80262046352314532635788470346, −3.59554687344549631003018020592, −3.31549814423213006283430285465, −3.03650784455289186903572829244, −2.75854425724953520010649723259, −1.91836667308977616396531829929, −1.90334398580523753014845680352, −1.61760611301596332199263100631, −0.801575136536248287435852033366, −0.66301121612278665301158155634, −0.44156962705337997371582218644,
0.44156962705337997371582218644, 0.66301121612278665301158155634, 0.801575136536248287435852033366, 1.61760611301596332199263100631, 1.90334398580523753014845680352, 1.91836667308977616396531829929, 2.75854425724953520010649723259, 3.03650784455289186903572829244, 3.31549814423213006283430285465, 3.59554687344549631003018020592, 3.80262046352314532635788470346, 3.99384385244426311562481846001, 4.53121116695540434977518117393, 4.60893112676358788196324467499, 4.90857428746292522172906084883, 5.12633724426551357687722013577, 5.45471184241227539161713615691, 5.56715452265414334711730264005, 5.95015214540643568874222890407, 6.16938542931036424871859811240, 6.24567174371663540764457455098, 6.69937079791830339775835528432, 6.76375131074506409994416460273, 7.14895817382775261480952973757, 7.47641027159507669913966469524
