| L(s) = 1 | − 3·3-s − 4-s + 4·7-s + 8-s + 2·9-s + 3·12-s − 9·13-s − 16-s − 2·17-s + 7·19-s − 12·21-s − 8·23-s − 3·24-s − 4·28-s + 11·29-s + 4·31-s − 2·32-s − 2·36-s + 6·37-s + 27·39-s − 5·41-s − 13·43-s − 24·47-s + 3·48-s + 6·51-s + 9·52-s − 23·53-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 1/2·4-s + 1.51·7-s + 0.353·8-s + 2/3·9-s + 0.866·12-s − 2.49·13-s − 1/4·16-s − 0.485·17-s + 1.60·19-s − 2.61·21-s − 1.66·23-s − 0.612·24-s − 0.755·28-s + 2.04·29-s + 0.718·31-s − 0.353·32-s − 1/3·36-s + 0.986·37-s + 4.32·39-s − 0.780·41-s − 1.98·43-s − 3.50·47-s + 0.433·48-s + 0.840·51-s + 1.24·52-s − 3.15·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{6} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{6} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | 5 | | \( 1 \) | |
| 11 | | \( 1 \) | |
| good | 2 | $S_4\times C_2$ | \( 1 + T^{2} - T^{3} + p T^{4} + p^{3} T^{6} \) | 3.2.a_b_ab |
| 3 | $S_4\times C_2$ | \( 1 + p T + 7 T^{2} + 5 p T^{3} + 7 p T^{4} + p^{3} T^{5} + p^{3} T^{6} \) | 3.3.d_h_p |
| 7 | $S_4\times C_2$ | \( 1 - 4 T + 16 T^{2} - 47 T^{3} + 16 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.7.ae_q_abv |
| 13 | $S_4\times C_2$ | \( 1 + 9 T + 61 T^{2} + 19 p T^{3} + 61 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.j_cj_jn |
| 17 | $S_4\times C_2$ | \( 1 + 2 T + 20 T^{2} - 7 T^{3} + 20 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.c_u_ah |
| 19 | $S_4\times C_2$ | \( 1 - 7 T + 63 T^{2} - 257 T^{3} + 63 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.ah_cl_ajx |
| 23 | $S_4\times C_2$ | \( 1 + 8 T + 58 T^{2} + 247 T^{3} + 58 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.i_cg_jn |
| 29 | $S_4\times C_2$ | \( 1 - 11 T + 119 T^{2} - 665 T^{3} + 119 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.al_ep_azp |
| 31 | $S_4\times C_2$ | \( 1 - 4 T + 57 T^{2} - 176 T^{3} + 57 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.ae_cf_agu |
| 37 | $S_4\times C_2$ | \( 1 - 6 T + 118 T^{2} - 443 T^{3} + 118 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.ag_eo_arb |
| 41 | $S_4\times C_2$ | \( 1 + 5 T + 90 T^{2} + 365 T^{3} + 90 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.f_dm_ob |
| 43 | $S_4\times C_2$ | \( 1 + 13 T + 48 T^{2} - 7 T^{3} + 48 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.n_bw_ah |
| 47 | $S_4\times C_2$ | \( 1 + 24 T + 328 T^{2} + 2727 T^{3} + 328 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.y_mq_eax |
| 53 | $S_4\times C_2$ | \( 1 + 23 T + 319 T^{2} + 2749 T^{3} + 319 p T^{4} + 23 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.x_mh_ebt |
| 59 | $S_4\times C_2$ | \( 1 + 13 T + 225 T^{2} + 1571 T^{3} + 225 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.n_ir_cil |
| 61 | $S_4\times C_2$ | \( 1 + 21 T + 325 T^{2} + 2869 T^{3} + 325 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.v_mn_egj |
| 67 | $S_4\times C_2$ | \( 1 - 10 T + 200 T^{2} - 1339 T^{3} + 200 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.ak_hs_abzn |
| 71 | $S_4\times C_2$ | \( 1 + 4 T + 118 T^{2} + 779 T^{3} + 118 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.e_eo_bdz |
| 73 | $S_4\times C_2$ | \( 1 + 3 T + 145 T^{2} + 241 T^{3} + 145 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.d_fp_jh |
| 79 | $S_4\times C_2$ | \( 1 + 16 T + 194 T^{2} + 1453 T^{3} + 194 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.q_hm_cdx |
| 83 | $S_4\times C_2$ | \( 1 - T + 21 T^{2} - 11 p T^{3} + 21 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) | 3.83.ab_v_abjd |
| 89 | $S_4\times C_2$ | \( 1 - 15 T + 253 T^{2} - 2477 T^{3} + 253 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.ap_jt_adrh |
| 97 | $S_4\times C_2$ | \( 1 + 6 T + 190 T^{2} + 871 T^{3} + 190 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.g_hi_bhn |
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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
−7.929270393972899824729040898512, −7.87787458599553965344438313555, −7.83509074359251535988694469870, −7.35554191687690164985888015480, −7.13557090726317441088559783286, −6.66135165857437862782394306960, −6.48517294209935754002965014394, −6.36147040226124165147839820798, −6.09247907480869997669309557190, −5.82438998505406899681502598588, −5.21981576335177316900999601974, −5.18921912173993582411339336494, −5.05595944468701090649466744261, −4.77024877454006157550728520442, −4.55758780829106875732677401037, −4.52479175422061036357864812412, −4.19421733119203839626740530248, −3.42550432302856239171241615870, −3.24550426374193067429739290084, −2.85072526700374314313083988945, −2.75616605756615398134406378290, −2.02698444958926279723818017185, −1.62832822328511787461735388321, −1.62259966048911427526674666697, −1.10366014998179732918885785265, 0, 0, 0,
1.10366014998179732918885785265, 1.62259966048911427526674666697, 1.62832822328511787461735388321, 2.02698444958926279723818017185, 2.75616605756615398134406378290, 2.85072526700374314313083988945, 3.24550426374193067429739290084, 3.42550432302856239171241615870, 4.19421733119203839626740530248, 4.52479175422061036357864812412, 4.55758780829106875732677401037, 4.77024877454006157550728520442, 5.05595944468701090649466744261, 5.18921912173993582411339336494, 5.21981576335177316900999601974, 5.82438998505406899681502598588, 6.09247907480869997669309557190, 6.36147040226124165147839820798, 6.48517294209935754002965014394, 6.66135165857437862782394306960, 7.13557090726317441088559783286, 7.35554191687690164985888015480, 7.83509074359251535988694469870, 7.87787458599553965344438313555, 7.929270393972899824729040898512
