| L(s) = 1 | − 2·2-s + 3-s + 3·4-s − 5-s − 2·6-s − 3·7-s − 4·8-s + 2·10-s − 3·11-s + 3·12-s + 6·14-s − 15-s + 3·16-s + 6·17-s − 16·19-s − 3·20-s − 3·21-s + 6·22-s − 5·23-s − 4·24-s − 6·25-s + 27-s − 9·28-s + 16·29-s + 2·30-s + 7·31-s − 6·32-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.577·3-s + 3/2·4-s − 0.447·5-s − 0.816·6-s − 1.13·7-s − 1.41·8-s + 0.632·10-s − 0.904·11-s + 0.866·12-s + 1.60·14-s − 0.258·15-s + 3/4·16-s + 1.45·17-s − 3.67·19-s − 0.670·20-s − 0.654·21-s + 1.27·22-s − 1.04·23-s − 0.816·24-s − 6/5·25-s + 0.192·27-s − 1.70·28-s + 2.97·29-s + 0.365·30-s + 1.25·31-s − 1.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{3} \cdot 11^{3} \cdot 41^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{3} \cdot 11^{3} \cdot 41^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 7 | $C_1$ | \( ( 1 + T )^{3} \) | |
| 11 | $C_1$ | \( ( 1 + T )^{3} \) | |
| 41 | $C_1$ | \( ( 1 + T )^{3} \) | |
| good | 2 | $D_{6}$ | \( 1 + p T + T^{2} + p T^{4} + p^{3} T^{5} + p^{3} T^{6} \) | 3.2.c_b_a |
| 3 | $S_4\times C_2$ | \( 1 - T + T^{2} - 2 T^{3} + p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) | 3.3.ab_b_ac |
| 5 | $S_4\times C_2$ | \( 1 + T + 7 T^{2} + 6 T^{3} + 7 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) | 3.5.b_h_g |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) | 3.13.a_bn_a |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{3} \) | 3.17.ag_cl_aie |
| 19 | $S_4\times C_2$ | \( 1 + 16 T + 134 T^{2} + 714 T^{3} + 134 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.q_fe_bbm |
| 23 | $S_4\times C_2$ | \( 1 + 5 T + 3 p T^{2} + 222 T^{3} + 3 p^{2} T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.f_cr_io |
| 29 | $S_4\times C_2$ | \( 1 - 16 T + 166 T^{2} - 1050 T^{3} + 166 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.aq_gk_abok |
| 31 | $S_4\times C_2$ | \( 1 - 7 T + 58 T^{2} - 193 T^{3} + 58 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.ah_cg_ahl |
| 37 | $S_4\times C_2$ | \( 1 - 13 T + 134 T^{2} - 909 T^{3} + 134 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.an_fe_abiz |
| 43 | $S_4\times C_2$ | \( 1 + 14 T + 188 T^{2} + 1272 T^{3} + 188 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.o_hg_bwy |
| 47 | $S_4\times C_2$ | \( 1 + 20 T + 241 T^{2} + 1944 T^{3} + 241 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.u_jh_cwu |
| 53 | $S_4\times C_2$ | \( 1 + 22 T + 295 T^{2} + 2508 T^{3} + 295 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.w_lj_dsm |
| 59 | $S_4\times C_2$ | \( 1 + 13 T + 209 T^{2} + 1538 T^{3} + 209 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.n_ib_che |
| 61 | $S_4\times C_2$ | \( 1 - 8 T + 76 T^{2} - 98 T^{3} + 76 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.ai_cy_adu |
| 67 | $S_4\times C_2$ | \( 1 + 17 T + 272 T^{2} + 2349 T^{3} + 272 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.r_km_dmj |
| 71 | $S_4\times C_2$ | \( 1 + 5 T + 124 T^{2} + 777 T^{3} + 124 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.f_eu_bdx |
| 73 | $S_4\times C_2$ | \( 1 + 12 T + 176 T^{2} + 1118 T^{3} + 176 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.m_gu_bra |
| 79 | $S_4\times C_2$ | \( 1 - 2 T + 105 T^{2} - 308 T^{3} + 105 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.ac_eb_alw |
| 83 | $S_4\times C_2$ | \( 1 + 16 T + 237 T^{2} + 2064 T^{3} + 237 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.q_jd_dbk |
| 89 | $S_4\times C_2$ | \( 1 - 15 T + 252 T^{2} - 2129 T^{3} + 252 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.ap_js_addx |
| 97 | $S_4\times C_2$ | \( 1 + 3 T + 284 T^{2} + 565 T^{3} + 284 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.d_ky_vt |
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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
−8.242978864167221602524233334268, −8.071771253799826383387637488961, −7.65704081206006856431928833718, −7.44306485441858582528459761926, −7.21912440991618710785383740121, −6.69756162750419690528555627607, −6.59431544715783715005653366066, −6.36491325839219147020317297451, −6.06555781054074320306759285959, −6.02908846996091882950993030612, −5.97596116315719577411053196921, −5.14739229434675988023156253009, −4.75452646622126637588706528937, −4.72811085227998696770640872487, −4.37501164316624856694143664583, −4.21693083473572188327330445888, −3.55543262984815049974953005608, −3.42478205037754111564264475233, −3.00451668299324360146202773259, −2.91035804539070084335278649982, −2.62451764013338462537629949545, −2.05179017410549886534511528235, −2.00792252691374856912312708411, −1.44106118905182855349122047244, −1.09360317117248128521157497219, 0, 0, 0,
1.09360317117248128521157497219, 1.44106118905182855349122047244, 2.00792252691374856912312708411, 2.05179017410549886534511528235, 2.62451764013338462537629949545, 2.91035804539070084335278649982, 3.00451668299324360146202773259, 3.42478205037754111564264475233, 3.55543262984815049974953005608, 4.21693083473572188327330445888, 4.37501164316624856694143664583, 4.72811085227998696770640872487, 4.75452646622126637588706528937, 5.14739229434675988023156253009, 5.97596116315719577411053196921, 6.02908846996091882950993030612, 6.06555781054074320306759285959, 6.36491325839219147020317297451, 6.59431544715783715005653366066, 6.69756162750419690528555627607, 7.21912440991618710785383740121, 7.44306485441858582528459761926, 7.65704081206006856431928833718, 8.071771253799826383387637488961, 8.242978864167221602524233334268
