| L(s) = 1 | − 3·2-s + 6·4-s + 3·5-s − 10·8-s − 9-s − 9·10-s − 3·11-s − 12·13-s + 15·16-s − 8·17-s + 3·18-s + 2·19-s + 18·20-s + 9·22-s + 6·25-s + 36·26-s + 5·27-s + 11·29-s + 5·31-s − 21·32-s + 24·34-s − 6·36-s − 7·37-s − 6·38-s − 30·40-s + 6·41-s + 3·43-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 3·4-s + 1.34·5-s − 3.53·8-s − 1/3·9-s − 2.84·10-s − 0.904·11-s − 3.32·13-s + 15/4·16-s − 1.94·17-s + 0.707·18-s + 0.458·19-s + 4.02·20-s + 1.91·22-s + 6/5·25-s + 7.06·26-s + 0.962·27-s + 2.04·29-s + 0.898·31-s − 3.71·32-s + 4.11·34-s − 36-s − 1.15·37-s − 0.973·38-s − 4.74·40-s + 0.937·41-s + 0.457·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 5^{3} \cdot 11^{3} \cdot 43^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 5^{3} \cdot 11^{3} \cdot 43^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{3} \) |
| 11 | $C_1$ | \( ( 1 + T )^{3} \) |
| 43 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 + T^{2} - 5 T^{3} + p T^{4} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 13 T^{2} + 5 T^{3} + 13 p T^{4} + p^{3} T^{6} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{3} \) |
| 17 | $S_4\times C_2$ | \( 1 + 8 T + 49 T^{2} + 191 T^{3} + 49 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 2 T + 37 T^{2} - 75 T^{3} + 37 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) |
| 29 | $S_4\times C_2$ | \( 1 - 11 T + 113 T^{2} - 650 T^{3} + 113 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 5 T + 57 T^{2} - 166 T^{3} + 57 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 7 T + 79 T^{2} + 534 T^{3} + 79 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 6 T + 103 T^{2} - 396 T^{3} + 103 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + T + 118 T^{2} + 124 T^{3} + 118 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 25 T^{2} - 597 T^{3} + 25 p T^{4} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 13 T + 212 T^{2} + 1510 T^{3} + 212 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - T + 45 T^{2} - 702 T^{3} + 45 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{3} \) |
| 71 | $S_4\times C_2$ | \( 1 + 12 T + 253 T^{2} + 1731 T^{3} + 253 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 28 T + 387 T^{2} + 3728 T^{3} + 387 p T^{4} + 28 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 3 T + 100 T^{2} - 274 T^{3} + 100 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 15 T + 184 T^{2} + 1854 T^{3} + 184 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - T + 121 T^{2} - 358 T^{3} + 121 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 20 T + 367 T^{2} - 3672 T^{3} + 367 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.79332009228627192991310111494, −7.29147707161509643710621578504, −7.27280298345585766024471790017, −7.22106967775464872229526263279, −6.86440263462481472875941559094, −6.56513465389446631775082640370, −6.41550383326852884831932655877, −6.05284211912560284508392623198, −5.82358555814069797750582019955, −5.80949262189882117135040353764, −5.07509958680007234685186318475, −5.01048508854634556697314196889, −4.77689083459532606200677268968, −4.67201107479405580394979476810, −4.39454791430195668527176140017, −3.89309999731543706146947659273, −3.15375378047847111496397409835, −2.94739732753632036215248803871, −2.80279247645922705865984349970, −2.54670416712544307134125563878, −2.48941168702190012581220229182, −2.05530232524308086717305109402, −1.72813143680326724805799009293, −1.18800944201495212580106351437, −1.16234284246933171883074792520, 0, 0, 0,
1.16234284246933171883074792520, 1.18800944201495212580106351437, 1.72813143680326724805799009293, 2.05530232524308086717305109402, 2.48941168702190012581220229182, 2.54670416712544307134125563878, 2.80279247645922705865984349970, 2.94739732753632036215248803871, 3.15375378047847111496397409835, 3.89309999731543706146947659273, 4.39454791430195668527176140017, 4.67201107479405580394979476810, 4.77689083459532606200677268968, 5.01048508854634556697314196889, 5.07509958680007234685186318475, 5.80949262189882117135040353764, 5.82358555814069797750582019955, 6.05284211912560284508392623198, 6.41550383326852884831932655877, 6.56513465389446631775082640370, 6.86440263462481472875941559094, 7.22106967775464872229526263279, 7.27280298345585766024471790017, 7.29147707161509643710621578504, 7.79332009228627192991310111494
