| L(s) = 1 | + 15·5-s − 5·7-s − 33·11-s + 2·13-s − 77·17-s + 171·19-s − 222·23-s + 150·25-s − 55·29-s + 181·31-s − 75·35-s + 317·37-s − 302·41-s − 188·43-s − 662·47-s − 636·49-s − 81·53-s − 495·55-s + 42·59-s + 349·61-s + 30·65-s + 152·67-s − 927·71-s − 2.37e3·73-s + 165·77-s − 1.14e3·79-s + 458·83-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 0.269·7-s − 0.904·11-s + 0.0426·13-s − 1.09·17-s + 2.06·19-s − 2.01·23-s + 6/5·25-s − 0.352·29-s + 1.04·31-s − 0.362·35-s + 1.40·37-s − 1.15·41-s − 0.666·43-s − 2.05·47-s − 1.85·49-s − 0.209·53-s − 1.21·55-s + 0.0926·59-s + 0.732·61-s + 0.0572·65-s + 0.277·67-s − 1.54·71-s − 3.81·73-s + 0.244·77-s − 1.63·79-s + 0.605·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 5^{3} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 5^{3} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - p T )^{3} \) |
| 11 | $C_1$ | \( ( 1 + p T )^{3} \) |
| good | 7 | $S_4\times C_2$ | \( 1 + 5 T + 661 T^{2} + 4330 T^{3} + 661 p^{3} T^{4} + 5 p^{6} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 2 T + 275 p T^{2} + 56572 T^{3} + 275 p^{4} T^{4} - 2 p^{6} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 77 T + 8811 T^{2} + 301106 T^{3} + 8811 p^{3} T^{4} + 77 p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 9 p T + 23361 T^{2} - 2299442 T^{3} + 23361 p^{3} T^{4} - 9 p^{7} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 222 T + 39405 T^{2} + 4353684 T^{3} + 39405 p^{3} T^{4} + 222 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 55 T + 10047 T^{2} + 1179922 T^{3} + 10047 p^{3} T^{4} + 55 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 181 T + 4669 T^{2} - 289622 T^{3} + 4669 p^{3} T^{4} - 181 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 317 T + 2423 T^{2} + 13714762 T^{3} + 2423 p^{3} T^{4} - 317 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 302 T + 188007 T^{2} + 37038404 T^{3} + 188007 p^{3} T^{4} + 302 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 188 T - 2627 T^{2} - 22061432 T^{3} - 2627 p^{3} T^{4} + 188 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 662 T + 403173 T^{2} + 134029892 T^{3} + 403173 p^{3} T^{4} + 662 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 81 T + 209031 T^{2} - 23296626 T^{3} + 209031 p^{3} T^{4} + 81 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 42 T + 304329 T^{2} - 63529692 T^{3} + 304329 p^{3} T^{4} - 42 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 349 T + 543655 T^{2} - 114299942 T^{3} + 543655 p^{3} T^{4} - 349 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 152 T + 671657 T^{2} - 122079056 T^{3} + 671657 p^{3} T^{4} - 152 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 927 T + 1008933 T^{2} + 560360466 T^{3} + 1008933 p^{3} T^{4} + 927 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 2378 T + 2919235 T^{2} + 2224720996 T^{3} + 2919235 p^{3} T^{4} + 2378 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 1146 T + 1780077 T^{2} + 1144096492 T^{3} + 1780077 p^{3} T^{4} + 1146 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 458 T + 868029 T^{2} - 619632500 T^{3} + 868029 p^{3} T^{4} - 458 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 875 T + 1992987 T^{2} - 1237529186 T^{3} + 1992987 p^{3} T^{4} - 875 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 1980 T + 2284359 T^{2} + 1843687192 T^{3} + 2284359 p^{3} T^{4} + 1980 p^{6} T^{5} + p^{9} T^{6} \) |
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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
−8.322655631732278776684654117235, −7.81956305323851466480556071028, −7.63143094375002394813061106697, −7.59878325497054720885523079830, −6.85541106266732728685204277486, −6.77822036213998269621735882055, −6.77525141590948760279077707789, −6.07906676719608203561187200263, −6.02558546537805123402680264274, −5.98033286199598182172823606730, −5.34516809294897388110364552402, −5.20733148184024751537342269989, −5.18615355190058449327157818518, −4.41054081362844691630161973731, −4.37570176425143219517414897881, −4.34319896692653058866882944189, −3.40065234105141545109787307436, −3.35393722466045073783809590721, −3.10474085084156910953948591395, −2.65975106462860878265485683118, −2.31218502970926064636229307912, −2.18697331931740785536370423127, −1.45731257573750951228962785689, −1.36157539911879044131951258031, −1.18269626910709040665353313841, 0, 0, 0,
1.18269626910709040665353313841, 1.36157539911879044131951258031, 1.45731257573750951228962785689, 2.18697331931740785536370423127, 2.31218502970926064636229307912, 2.65975106462860878265485683118, 3.10474085084156910953948591395, 3.35393722466045073783809590721, 3.40065234105141545109787307436, 4.34319896692653058866882944189, 4.37570176425143219517414897881, 4.41054081362844691630161973731, 5.18615355190058449327157818518, 5.20733148184024751537342269989, 5.34516809294897388110364552402, 5.98033286199598182172823606730, 6.02558546537805123402680264274, 6.07906676719608203561187200263, 6.77525141590948760279077707789, 6.77822036213998269621735882055, 6.85541106266732728685204277486, 7.59878325497054720885523079830, 7.63143094375002394813061106697, 7.81956305323851466480556071028, 8.322655631732278776684654117235
