| L(s) = 1 | + 3·2-s + 3·3-s + 6·4-s − 3·5-s + 9·6-s + 7-s + 10·8-s + 6·9-s − 9·10-s − 3·11-s + 18·12-s + 5·13-s + 3·14-s − 9·15-s + 15·16-s + 3·17-s + 18·18-s + 13·19-s − 18·20-s + 3·21-s − 9·22-s − 11·23-s + 30·24-s + 6·25-s + 15·26-s + 10·27-s + 6·28-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 1.73·3-s + 3·4-s − 1.34·5-s + 3.67·6-s + 0.377·7-s + 3.53·8-s + 2·9-s − 2.84·10-s − 0.904·11-s + 5.19·12-s + 1.38·13-s + 0.801·14-s − 2.32·15-s + 15/4·16-s + 0.727·17-s + 4.24·18-s + 2.98·19-s − 4.02·20-s + 0.654·21-s − 1.91·22-s − 2.29·23-s + 6.12·24-s + 6/5·25-s + 2.94·26-s + 1.92·27-s + 1.13·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 5^{3} \cdot 11^{3} \cdot 17^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 5^{3} \cdot 11^{3} \cdot 17^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(58.94913195\) |
| \(L(\frac12)\) |
\(\approx\) |
\(58.94913195\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{3} \) |
| 3 | $C_1$ | \( ( 1 - T )^{3} \) |
| 5 | $C_1$ | \( ( 1 + T )^{3} \) |
| 11 | $C_1$ | \( ( 1 + T )^{3} \) |
| 17 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 7 | $S_4\times C_2$ | \( 1 - T + p T^{2} + 2 T^{3} + p^{2} T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 5 T + 33 T^{2} - 122 T^{3} + 33 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 13 T + 99 T^{2} - 502 T^{3} + 99 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 11 T + 95 T^{2} + 514 T^{3} + 95 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{3} \) |
| 31 | $S_4\times C_2$ | \( 1 - 5 T + 21 T^{2} + 58 T^{3} + 21 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - T + 59 T^{2} - 198 T^{3} + 59 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{3} \) |
| 43 | $S_4\times C_2$ | \( 1 - 6 T + 89 T^{2} - 388 T^{3} + 89 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 10 T - 35 T^{2} + 852 T^{3} - 35 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 12 T + 155 T^{2} - 1160 T^{3} + 155 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 6 T + 137 T^{2} - 580 T^{3} + 137 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 11 T + 171 T^{2} - 1058 T^{3} + 171 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 29 T + 429 T^{2} - 4142 T^{3} + 429 p T^{4} - 29 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 4 T + 137 T^{2} - 440 T^{3} + 137 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 10 T + 115 T^{2} - 436 T^{3} + 115 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 8 T + 177 T^{2} - 1088 T^{3} + 177 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 19 T + 289 T^{2} + 3138 T^{3} + 289 p T^{4} + 19 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 2 T - 57 T^{2} - 36 T^{3} - 57 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 9 T + 95 T^{2} - 110 T^{3} + 95 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.31220807175562205223419265095, −7.06088387201508506871747512798, −6.65889073274200544270750598425, −6.59292266892325786285659398431, −6.07929651214859143691561899186, −6.00688541238882429880225040201, −5.55985628675021215554984245715, −5.32402848606899467314538567341, −5.31287241451343350324195641174, −4.96972523711986785022394533978, −4.58444377535302019798648341882, −4.42036035790833188444687031653, −4.01137742722581749215115606645, −3.81958715554762427612584031141, −3.75784329859456685803801349991, −3.55702019419960767748169858130, −3.04543220211847221757082554308, −3.02782274524639524396477328819, −2.89048129332289778999117090659, −2.19869413718777123405851256781, −2.06954568093964844056573119252, −2.05030075191844266794782363152, −1.06384498742223717873739093744, −0.989786910029940653778111025198, −0.817902698665084846848718219766,
0.817902698665084846848718219766, 0.989786910029940653778111025198, 1.06384498742223717873739093744, 2.05030075191844266794782363152, 2.06954568093964844056573119252, 2.19869413718777123405851256781, 2.89048129332289778999117090659, 3.02782274524639524396477328819, 3.04543220211847221757082554308, 3.55702019419960767748169858130, 3.75784329859456685803801349991, 3.81958715554762427612584031141, 4.01137742722581749215115606645, 4.42036035790833188444687031653, 4.58444377535302019798648341882, 4.96972523711986785022394533978, 5.31287241451343350324195641174, 5.32402848606899467314538567341, 5.55985628675021215554984245715, 6.00688541238882429880225040201, 6.07929651214859143691561899186, 6.59292266892325786285659398431, 6.65889073274200544270750598425, 7.06088387201508506871747512798, 7.31220807175562205223419265095
