| L(s) = 1 | + 3·5-s − 3·9-s + 6·11-s + 6·23-s + 6·25-s + 2·27-s − 6·29-s + 6·31-s + 12·37-s − 6·41-s − 6·43-s − 9·45-s + 12·47-s − 9·49-s − 6·53-s + 18·55-s − 24·59-s − 6·61-s − 12·67-s + 24·73-s − 12·79-s + 12·83-s + 24·89-s + 18·97-s − 18·99-s − 6·101-s − 12·103-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 9-s + 1.80·11-s + 1.25·23-s + 6/5·25-s + 0.384·27-s − 1.11·29-s + 1.07·31-s + 1.97·37-s − 0.937·41-s − 0.914·43-s − 1.34·45-s + 1.75·47-s − 9/7·49-s − 0.824·53-s + 2.42·55-s − 3.12·59-s − 0.768·61-s − 1.46·67-s + 2.80·73-s − 1.35·79-s + 1.31·83-s + 2.54·89-s + 1.82·97-s − 1.80·99-s − 0.597·101-s − 1.18·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.454127176\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.454127176\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{3} \) |
| 13 | | \( 1 \) |
| good | 3 | $S_4\times C_2$ | \( 1 + p T^{2} - 2 T^{3} + p^{2} T^{4} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 9 T^{2} + 12 T^{3} + 9 p T^{4} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 6 T + 3 p T^{2} - 114 T^{3} + 3 p^{2} T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) |
| 19 | $S_4\times C_2$ | \( 1 + 9 T^{2} + 6 p T^{3} + 9 p T^{4} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 6 T + 39 T^{2} - 102 T^{3} + 39 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 6 T + 75 T^{2} + 264 T^{3} + 75 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 6 T + 81 T^{2} - 306 T^{3} + 81 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 12 T + 3 p T^{2} - 636 T^{3} + 3 p^{2} T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 6 T + 87 T^{2} + 420 T^{3} + 87 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 6 T + 87 T^{2} + 470 T^{3} + 87 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 12 T + 105 T^{2} - 660 T^{3} + 105 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 6 T + 75 T^{2} + 660 T^{3} + 75 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 24 T + 357 T^{2} + 3246 T^{3} + 357 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 6 T + 87 T^{2} + 200 T^{3} + 87 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 12 T + 153 T^{2} + 1020 T^{3} + 153 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 3 T^{2} - 702 T^{3} - 3 p T^{4} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 24 T + 363 T^{2} - 3756 T^{3} + 363 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 12 T + 93 T^{2} + 200 T^{3} + 93 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 12 T + 3 p T^{2} - 1740 T^{3} + 3 p^{2} T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 24 T + 3 p T^{2} - 2256 T^{3} + 3 p^{2} T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 18 T + 351 T^{2} - 3516 T^{3} + 351 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.80382528809538244941006990293, −7.41391524767570049627124251575, −7.02625911624779116482463573803, −6.69711875150671493787951276820, −6.56095157822754553106056105817, −6.37825050974394212102978535251, −6.12798213734268256308011443653, −5.86227129635290645490742037074, −5.79071218668798591761283333036, −5.40823499143385000683210246605, −4.94127986577809703582793239276, −4.75297502228793133709639180381, −4.60215433537636180782343378281, −4.45079067812722091913498550190, −3.75844589923514895353752836833, −3.68381993221106659668510947671, −3.07341322162719367414542733606, −3.01333361736635170455142801204, −3.00471564472374930530643431648, −2.23195073069618392004546578871, −1.98809386082915146684609802538, −1.72689212719307816468248807149, −1.36103245414624718849959780079, −0.806781870174950945219295232268, −0.57715431274791175218619967920,
0.57715431274791175218619967920, 0.806781870174950945219295232268, 1.36103245414624718849959780079, 1.72689212719307816468248807149, 1.98809386082915146684609802538, 2.23195073069618392004546578871, 3.00471564472374930530643431648, 3.01333361736635170455142801204, 3.07341322162719367414542733606, 3.68381993221106659668510947671, 3.75844589923514895353752836833, 4.45079067812722091913498550190, 4.60215433537636180782343378281, 4.75297502228793133709639180381, 4.94127986577809703582793239276, 5.40823499143385000683210246605, 5.79071218668798591761283333036, 5.86227129635290645490742037074, 6.12798213734268256308011443653, 6.37825050974394212102978535251, 6.56095157822754553106056105817, 6.69711875150671493787951276820, 7.02625911624779116482463573803, 7.41391524767570049627124251575, 7.80382528809538244941006990293
