| L(s) = 1 | + 2-s + 3-s − 3·5-s + 6-s + 7-s − 3·10-s − 11-s + 14-s − 3·15-s + 21-s − 22-s + 6·25-s − 3·30-s − 31-s − 33-s − 3·35-s + 37-s − 41-s + 42-s − 3·43-s + 6·50-s + 3·55-s − 59-s − 62-s − 66-s − 3·70-s + 73-s + ⋯ |
| L(s) = 1 | + 2-s + 3-s − 3·5-s + 6-s + 7-s − 3·10-s − 11-s + 14-s − 3·15-s + 21-s − 22-s + 6·25-s − 3·30-s − 31-s − 33-s − 3·35-s + 37-s − 41-s + 42-s − 3·43-s + 6·50-s + 3·55-s − 59-s − 62-s − 66-s − 3·70-s + 73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9938375 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9938375 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4816455828\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4816455828\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_1$ | \( ( 1 + T )^{3} \) |
| 43 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 2 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 3 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 7 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 11 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 31 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 37 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 41 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 59 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 73 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 79 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
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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
−11.34379125328785296403961649266, −11.22519066620077210182278220712, −10.76565789894504762161654787937, −10.41951414670478913632550092101, −10.22614470519532324954738255252, −9.451598184601566049993776331296, −9.210453815185567297377928012003, −8.562767054972417158033702296369, −8.561025692932832088038138717078, −8.061423278316292225110118443792, −7.965802995817511320929650348032, −7.87365244740645662689143737302, −7.14018871624886756240152041229, −7.05628714597172025117836233810, −6.63876829149795493887275500001, −5.91120385812736076212260808885, −5.11791066952921739132242545141, −5.06578404007996484477578285838, −4.69094714740688285600725508569, −4.34881257171637828770630153402, −3.83118423851181836123848265935, −3.59603625235340944718523456014, −2.97959148365389733494468329097, −2.83322874116644630311704377186, −1.69015443490494411863181296484,
1.69015443490494411863181296484, 2.83322874116644630311704377186, 2.97959148365389733494468329097, 3.59603625235340944718523456014, 3.83118423851181836123848265935, 4.34881257171637828770630153402, 4.69094714740688285600725508569, 5.06578404007996484477578285838, 5.11791066952921739132242545141, 5.91120385812736076212260808885, 6.63876829149795493887275500001, 7.05628714597172025117836233810, 7.14018871624886756240152041229, 7.87365244740645662689143737302, 7.965802995817511320929650348032, 8.061423278316292225110118443792, 8.561025692932832088038138717078, 8.562767054972417158033702296369, 9.210453815185567297377928012003, 9.451598184601566049993776331296, 10.22614470519532324954738255252, 10.41951414670478913632550092101, 10.76565789894504762161654787937, 11.22519066620077210182278220712, 11.34379125328785296403961649266
