| L(s) = 1 | − 2-s − 3-s − 5-s + 6-s + 10-s + 15-s − 19-s − 29-s − 30-s − 37-s + 38-s − 43-s + 3·49-s + 57-s + 58-s + 3·71-s − 73-s + 74-s − 79-s − 83-s + 86-s + 87-s − 89-s + 95-s − 3·98-s − 101-s − 103-s + ⋯ |
| L(s) = 1 | − 2-s − 3-s − 5-s + 6-s + 10-s + 15-s − 19-s − 29-s − 30-s − 37-s + 38-s − 43-s + 3·49-s + 57-s + 58-s + 3·71-s − 73-s + 74-s − 79-s − 83-s + 86-s + 87-s − 89-s + 95-s − 3·98-s − 101-s − 103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 357911 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 357911 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.05498223480\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05498223480\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 71 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 2 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 3 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 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
−13.58399519906890803499351778176, −12.88778415189161766638407934767, −12.62842911673974380337261507666, −12.42351194541329992426458232503, −11.75493220562099759167484344236, −11.67985104016781996899636064300, −11.17081547941659221637655464575, −11.05204135964904542931077288526, −10.37263061263202496602275828991, −10.24785144415084892602116520791, −9.646341880691708918339787590058, −9.249596659145116242975209696115, −8.775621347957351956553642935712, −8.392664296558968410521780245040, −8.308375254782647564178629435651, −7.34866153009380292102541584258, −7.32935458632589252277476848657, −6.76877928720481535930982673330, −6.02203245587172561884189386059, −5.76082408647750230290643880271, −5.19361966854786324344583978535, −4.51844117616935213608968885293, −3.95531641366854447656333489742, −3.39979849702102685652285544820, −2.19448912992332732423794638884,
2.19448912992332732423794638884, 3.39979849702102685652285544820, 3.95531641366854447656333489742, 4.51844117616935213608968885293, 5.19361966854786324344583978535, 5.76082408647750230290643880271, 6.02203245587172561884189386059, 6.76877928720481535930982673330, 7.32935458632589252277476848657, 7.34866153009380292102541584258, 8.308375254782647564178629435651, 8.392664296558968410521780245040, 8.775621347957351956553642935712, 9.249596659145116242975209696115, 9.646341880691708918339787590058, 10.24785144415084892602116520791, 10.37263061263202496602275828991, 11.05204135964904542931077288526, 11.17081547941659221637655464575, 11.67985104016781996899636064300, 11.75493220562099759167484344236, 12.42351194541329992426458232503, 12.62842911673974380337261507666, 12.88778415189161766638407934767, 13.58399519906890803499351778176
