| L(s) = 1 | − 3·3-s − 3·5-s + 2·7-s + 6·9-s − 2·11-s + 2·13-s + 9·15-s + 10·17-s − 6·21-s − 2·23-s + 6·25-s − 10·27-s + 6·29-s − 3·31-s + 6·33-s − 6·35-s + 4·37-s − 6·39-s + 16·41-s − 2·43-s − 18·45-s − 12·47-s − 11·49-s − 30·51-s + 6·53-s + 6·55-s − 16·59-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 1.34·5-s + 0.755·7-s + 2·9-s − 0.603·11-s + 0.554·13-s + 2.32·15-s + 2.42·17-s − 1.30·21-s − 0.417·23-s + 6/5·25-s − 1.92·27-s + 1.11·29-s − 0.538·31-s + 1.04·33-s − 1.01·35-s + 0.657·37-s − 0.960·39-s + 2.49·41-s − 0.304·43-s − 2.68·45-s − 1.75·47-s − 1.57·49-s − 4.20·51-s + 0.824·53-s + 0.809·55-s − 2.08·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 5^{3} \cdot 31^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 5^{3} \cdot 31^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.122681386\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.122681386\) |
| \(L(\frac{3}{2})\) |
|
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 | 2 | | \( 1 \) | |
| 3 | $C_1$ | \( ( 1 + T )^{3} \) | |
| 5 | $C_1$ | \( ( 1 + T )^{3} \) | |
| 31 | $C_1$ | \( ( 1 + T )^{3} \) | |
| good | 7 | $S_4\times C_2$ | \( 1 - 2 T + 15 T^{2} - 30 T^{3} + 15 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.7.ac_p_abe |
| 11 | $S_4\times C_2$ | \( 1 + 2 T + 5 T^{2} - 28 T^{3} + 5 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.c_f_abc |
| 13 | $S_4\times C_2$ | \( 1 - 2 T + 21 T^{2} - 70 T^{3} + 21 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.ac_v_acs |
| 17 | $S_4\times C_2$ | \( 1 - 10 T + 67 T^{2} - 304 T^{3} + 67 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.ak_cp_als |
| 19 | $S_4\times C_2$ | \( 1 + 25 T^{2} - 32 T^{3} + 25 p T^{4} + p^{3} T^{6} \) | 3.19.a_z_abg |
| 23 | $S_4\times C_2$ | \( 1 + 2 T + 29 T^{2} + 8 T^{3} + 29 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.c_bd_i |
| 29 | $S_4\times C_2$ | \( 1 - 6 T + 51 T^{2} - 186 T^{3} + 51 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.ag_bz_ahe |
| 37 | $S_4\times C_2$ | \( 1 - 4 T + 21 T^{2} + 150 T^{3} + 21 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.ae_v_fu |
| 41 | $S_4\times C_2$ | \( 1 - 16 T + 179 T^{2} - 1360 T^{3} + 179 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.aq_gx_acai |
| 43 | $S_4\times C_2$ | \( 1 + 2 T + 109 T^{2} + 180 T^{3} + 109 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.c_ef_gy |
| 47 | $S_4\times C_2$ | \( 1 + 12 T + 181 T^{2} + 1164 T^{3} + 181 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.m_gz_bsu |
| 53 | $S_4\times C_2$ | \( 1 - 6 T + 163 T^{2} - 624 T^{3} + 163 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.ag_gh_aya |
| 59 | $S_4\times C_2$ | \( 1 + 16 T + 257 T^{2} + 2014 T^{3} + 257 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.q_jx_czm |
| 61 | $S_4\times C_2$ | \( 1 - 14 T + 163 T^{2} - 1236 T^{3} + 163 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.ao_gh_abvo |
| 67 | $S_4\times C_2$ | \( 1 + 8 T + 127 T^{2} + 1158 T^{3} + 127 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.i_ex_bso |
| 71 | $S_4\times C_2$ | \( 1 + 10 T + 137 T^{2} + 1042 T^{3} + 137 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.k_fh_boc |
| 73 | $S_4\times C_2$ | \( 1 - 8 T + 145 T^{2} - 1254 T^{3} + 145 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.ai_fp_abwg |
| 79 | $S_4\times C_2$ | \( 1 - 16 T + 265 T^{2} - 2532 T^{3} + 265 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.aq_kf_adtk |
| 83 | $S_4\times C_2$ | \( 1 + 26 T + 457 T^{2} + 4832 T^{3} + 457 p T^{4} + 26 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.ba_rp_hdw |
| 89 | $S_4\times C_2$ | \( 1 + 8 T + 179 T^{2} + 1202 T^{3} + 179 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.i_gx_bug |
| 97 | $S_4\times C_2$ | \( 1 - 8 T + 283 T^{2} - 1440 T^{3} + 283 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.ai_kx_acdk |
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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
−7.14743018970984496659977866312, −6.55156689910090769139081628889, −6.48412376387746984260856639128, −6.38978354651703028003178721361, −5.86014814246004203172467438087, −5.78732787080664374223761083888, −5.65960946002533526550397483843, −5.22617414292820246563334613942, −5.06423856299574849262439279493, −5.04704167818703489715561821487, −4.46381266773060031124675643651, −4.42558099419097340603814272072, −4.18040331590792838638807817517, −3.98020038958256145164767647915, −3.56659943534486740911912054699, −3.32711579910307443639748951286, −3.07478615530304964810777717843, −2.70714749147347430585301884636, −2.61268676903421845063843017153, −1.83519392128849129347232209315, −1.52221798097605548899964067931, −1.47719596001949104158178328434, −0.874561776316442682954139683382, −0.73778774823554360209077541665, −0.26541054229516452713932751751,
0.26541054229516452713932751751, 0.73778774823554360209077541665, 0.874561776316442682954139683382, 1.47719596001949104158178328434, 1.52221798097605548899964067931, 1.83519392128849129347232209315, 2.61268676903421845063843017153, 2.70714749147347430585301884636, 3.07478615530304964810777717843, 3.32711579910307443639748951286, 3.56659943534486740911912054699, 3.98020038958256145164767647915, 4.18040331590792838638807817517, 4.42558099419097340603814272072, 4.46381266773060031124675643651, 5.04704167818703489715561821487, 5.06423856299574849262439279493, 5.22617414292820246563334613942, 5.65960946002533526550397483843, 5.78732787080664374223761083888, 5.86014814246004203172467438087, 6.38978354651703028003178721361, 6.48412376387746984260856639128, 6.55156689910090769139081628889, 7.14743018970984496659977866312
