L(s) = 1 | + 2-s + 5-s + 10-s − 19-s + 29-s − 37-s − 38-s − 43-s + 3·49-s + 58-s − 3·71-s − 73-s − 74-s − 79-s + 83-s − 86-s + 89-s − 95-s + 3·98-s + 101-s − 103-s − 6·107-s − 109-s + 3·121-s + 127-s − 128-s + 131-s + ⋯ |
L(s) = 1 | + 2-s + 5-s + 10-s − 19-s + 29-s − 37-s − 38-s − 43-s + 3·49-s + 58-s − 3·71-s − 73-s − 74-s − 79-s + 83-s − 86-s + 89-s − 95-s + 3·98-s + 101-s − 103-s − 6·107-s − 109-s + 3·121-s + 127-s − 128-s + 131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 71^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 71^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.330590273\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.330590273\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 3 | | \( 1 \) |
| 71 | $C_1$ | \( ( 1 + T )^{3} \) |
good | 2 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 5 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 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 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 29 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 37 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 43 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 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_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 67 | $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 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 89 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 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
−9.833420775873518817936336720201, −9.170241421950823689049236844172, −9.146777271039715882952195593566, −8.836934329516558101851519555806, −8.486252079886083166878407002878, −8.264653899015990838034506744415, −7.84966148329591903118429176352, −7.53479988776860593174990930549, −7.01065521827902659158644931458, −6.96469621423864126178713937736, −6.57025284206264928971284169702, −6.17296074601218658421437252310, −5.94983306036910194011291288850, −5.44912832043962092066970112486, −5.43620883403123476947121773662, −5.03167812273964799938864423182, −4.43567647761754511569812735999, −4.28360347967664439419897041164, −4.18420663369478119619908585192, −3.55182342283842946945061968422, −3.08842507256238758207915794763, −2.69641103570264554325485886409, −2.28751464686665026535167388429, −1.77073079791246240609274844577, −1.26584969481661945262580678838,
1.26584969481661945262580678838, 1.77073079791246240609274844577, 2.28751464686665026535167388429, 2.69641103570264554325485886409, 3.08842507256238758207915794763, 3.55182342283842946945061968422, 4.18420663369478119619908585192, 4.28360347967664439419897041164, 4.43567647761754511569812735999, 5.03167812273964799938864423182, 5.43620883403123476947121773662, 5.44912832043962092066970112486, 5.94983306036910194011291288850, 6.17296074601218658421437252310, 6.57025284206264928971284169702, 6.96469621423864126178713937736, 7.01065521827902659158644931458, 7.53479988776860593174990930549, 7.84966148329591903118429176352, 8.264653899015990838034506744415, 8.486252079886083166878407002878, 8.836934329516558101851519555806, 9.146777271039715882952195593566, 9.170241421950823689049236844172, 9.833420775873518817936336720201