| L(s) = 1 | + 3·2-s + 2·3-s + 6·4-s + 3·5-s + 6·6-s − 3·7-s + 10·8-s + 9-s + 9·10-s + 3·11-s + 12·12-s + 2·13-s − 9·14-s + 6·15-s + 15·16-s + 2·17-s + 3·18-s − 6·19-s + 18·20-s − 6·21-s + 9·22-s + 10·23-s + 20·24-s + 6·25-s + 6·26-s − 18·28-s + 18·30-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 1.15·3-s + 3·4-s + 1.34·5-s + 2.44·6-s − 1.13·7-s + 3.53·8-s + 1/3·9-s + 2.84·10-s + 0.904·11-s + 3.46·12-s + 0.554·13-s − 2.40·14-s + 1.54·15-s + 15/4·16-s + 0.485·17-s + 0.707·18-s − 1.37·19-s + 4.02·20-s − 1.30·21-s + 1.91·22-s + 2.08·23-s + 4.08·24-s + 6/5·25-s + 1.17·26-s − 3.40·28-s + 3.28·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 5^{3} \cdot 7^{3} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 5^{3} \cdot 7^{3} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(21.13508832\) |
| \(L(\frac12)\) |
\(\approx\) |
\(21.13508832\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{3} \) |
| 5 | $C_1$ | \( ( 1 - T )^{3} \) |
| 7 | $C_1$ | \( ( 1 + T )^{3} \) |
| 11 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 - 2 T + p T^{2} - 4 T^{3} + p^{2} T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 2 T + 23 T^{2} - 36 T^{3} + 23 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 2 T + 23 T^{2} - 76 T^{3} + 23 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 6 T + 11 T^{2} - 4 T^{3} + 11 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 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 | $S_4\times C_2$ | \( 1 + 21 T^{2} - 92 T^{3} + 21 p T^{4} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 12 T + 113 T^{2} + 712 T^{3} + 113 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 4 T + 109 T^{2} - 292 T^{3} + 109 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 4 T + 33 T^{2} - 12 p T^{3} + 33 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 8 T + 21 T^{2} + 160 T^{3} + 21 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 29 T^{2} - 128 T^{3} + 29 p T^{4} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 12 T + 141 T^{2} - 980 T^{3} + 141 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 4 T + 113 T^{2} + 344 T^{3} + 113 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 6 T + 83 T^{2} + 388 T^{3} + 83 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 4 T + 89 T^{2} - 600 T^{3} + 89 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 28 T + 413 T^{2} + 4040 T^{3} + 413 p T^{4} + 28 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 14 T + 255 T^{2} + 2036 T^{3} + 255 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 18 T + 287 T^{2} - 2588 T^{3} + 287 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 8 T + 241 T^{2} + 1296 T^{3} + 241 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 18 T + 239 T^{2} - 1996 T^{3} + 239 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 4 T + 137 T^{2} + 1092 T^{3} + 137 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−9.194817351196831512690578516945, −8.858451861415374889270035817489, −8.680535259474458192668156599372, −8.660238336340254391117015878922, −7.73110618060552022528735855423, −7.62554813657593880164831646119, −7.31777770190098032370246379843, −7.00127292544751390939920300063, −6.63584654406059732046915504691, −6.50584601323233142255906927546, −5.95234927606692060360042708197, −5.88937226039169312449635433005, −5.79581028823222080102084912225, −5.17535889645264415655106091807, −4.95689128401345261924760801554, −4.47881869210866016638093045778, −4.07148139226543421518082482763, −3.85259624764558711104415686920, −3.53138046693611425161815401095, −3.04070474583460291639805908798, −2.81670410459936213519775161119, −2.56862858462667886089264267108, −2.07917122607290327788046218850, −1.56786654656033659953808894510, −1.09760716782707600895209307056,
1.09760716782707600895209307056, 1.56786654656033659953808894510, 2.07917122607290327788046218850, 2.56862858462667886089264267108, 2.81670410459936213519775161119, 3.04070474583460291639805908798, 3.53138046693611425161815401095, 3.85259624764558711104415686920, 4.07148139226543421518082482763, 4.47881869210866016638093045778, 4.95689128401345261924760801554, 5.17535889645264415655106091807, 5.79581028823222080102084912225, 5.88937226039169312449635433005, 5.95234927606692060360042708197, 6.50584601323233142255906927546, 6.63584654406059732046915504691, 7.00127292544751390939920300063, 7.31777770190098032370246379843, 7.62554813657593880164831646119, 7.73110618060552022528735855423, 8.660238336340254391117015878922, 8.680535259474458192668156599372, 8.858451861415374889270035817489, 9.194817351196831512690578516945
