| 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
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1967933382\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1967933382\) |
| \(L(1)\) |
|
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
−10.96489997040161089926265142834, −10.84458079810731533979836554990, −10.51422850658961158494579249534, −10.20771776588144417504393781484, −10.01592267987270268279681762982, −9.509839761488073309852584200133, −9.475915223012095239339879710220, −9.091346296910843740055462064598, −8.726778995670167242116127383123, −8.720030138269798928259119557193, −7.938347713487745106158437643482, −7.48877529383275850675022842587, −7.02209622273135713470076064732, −6.74813139109988508890263600823, −6.15950774428888067403723291008, −6.15635080810380935651287222914, −5.62066092155808642348522122392, −5.45599328091309236741064651773, −5.16452478388338397775971422287, −4.62587065332912961508517776705, −3.88173496377688101154092717999, −2.94542724868869697488546024209, −2.79800051161472241723134558819, −2.10957644260712257194177020233, −1.44074276504072907473523790026,
1.44074276504072907473523790026, 2.10957644260712257194177020233, 2.79800051161472241723134558819, 2.94542724868869697488546024209, 3.88173496377688101154092717999, 4.62587065332912961508517776705, 5.16452478388338397775971422287, 5.45599328091309236741064651773, 5.62066092155808642348522122392, 6.15635080810380935651287222914, 6.15950774428888067403723291008, 6.74813139109988508890263600823, 7.02209622273135713470076064732, 7.48877529383275850675022842587, 7.938347713487745106158437643482, 8.720030138269798928259119557193, 8.726778995670167242116127383123, 9.091346296910843740055462064598, 9.475915223012095239339879710220, 9.509839761488073309852584200133, 10.01592267987270268279681762982, 10.20771776588144417504393781484, 10.51422850658961158494579249534, 10.84458079810731533979836554990, 10.96489997040161089926265142834
