| L(s) = 1 | − 2·4-s + 4·7-s − 2·8-s + 7·11-s + 7·17-s − 11·19-s + 5·23-s − 8·28-s + 5·29-s − 3·31-s + 8·32-s + 26·37-s + 8·41-s + 8·43-s − 14·44-s − 10·47-s − 5·49-s + 11·53-s − 8·56-s + 14·59-s + 12·61-s + 4·64-s + 3·67-s − 14·68-s + 26·71-s + 12·73-s + 22·76-s + ⋯ |
| L(s) = 1 | − 4-s + 1.51·7-s − 0.707·8-s + 2.11·11-s + 1.69·17-s − 2.52·19-s + 1.04·23-s − 1.51·28-s + 0.928·29-s − 0.538·31-s + 1.41·32-s + 4.27·37-s + 1.24·41-s + 1.21·43-s − 2.11·44-s − 1.45·47-s − 5/7·49-s + 1.51·53-s − 1.06·56-s + 1.82·59-s + 1.53·61-s + 1/2·64-s + 0.366·67-s − 1.69·68-s + 3.08·71-s + 1.40·73-s + 2.52·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 31^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 31^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.565032749\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.565032749\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | | \( 1 \) | |
| 5 | | \( 1 \) | |
| 31 | $C_1$ | \( ( 1 + T )^{3} \) | |
| good | 2 | $S_4\times C_2$ | \( 1 + p T^{2} + p T^{3} + p^{2} T^{4} + p^{3} T^{6} \) | 3.2.a_c_c |
| 7 | $S_4\times C_2$ | \( 1 - 4 T + 3 p T^{2} - 52 T^{3} + 3 p^{2} T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.7.ae_v_aca |
| 11 | $S_4\times C_2$ | \( 1 - 7 T + 36 T^{2} - 117 T^{3} + 36 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.ah_bk_aen |
| 13 | $S_4\times C_2$ | \( 1 + 5 T^{2} + 62 T^{3} + 5 p T^{4} + p^{3} T^{6} \) | 3.13.a_f_ck |
| 17 | $S_4\times C_2$ | \( 1 - 7 T + 44 T^{2} - 179 T^{3} + 44 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.ah_bs_agx |
| 19 | $S_4\times C_2$ | \( 1 + 11 T + 84 T^{2} + 419 T^{3} + 84 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.l_dg_qd |
| 23 | $S_4\times C_2$ | \( 1 - 5 T + 12 T^{2} + 9 T^{3} + 12 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.af_m_j |
| 29 | $S_4\times C_2$ | \( 1 - 5 T + 92 T^{2} - 289 T^{3} + 92 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.af_do_ald |
| 37 | $S_4\times C_2$ | \( 1 - 26 T + 9 p T^{2} - 2546 T^{3} + 9 p^{2} T^{4} - 26 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.aba_mv_adty |
| 41 | $S_4\times C_2$ | \( 1 - 8 T + 107 T^{2} - 526 T^{3} + 107 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.ai_ed_aug |
| 43 | $S_4\times C_2$ | \( 1 - 8 T + 57 T^{2} - 320 T^{3} + 57 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.ai_cf_ami |
| 47 | $S_4\times C_2$ | \( 1 + 10 T + 121 T^{2} + 948 T^{3} + 121 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.k_er_bkm |
| 53 | $S_4\times C_2$ | \( 1 - 11 T + 120 T^{2} - 1191 T^{3} + 120 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.al_eq_abtv |
| 59 | $S_4\times C_2$ | \( 1 - 14 T + 181 T^{2} - 1350 T^{3} + 181 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.ao_gz_abzy |
| 61 | $S_4\times C_2$ | \( 1 - 12 T + 167 T^{2} - 1400 T^{3} + 167 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.am_gl_acbw |
| 67 | $S_4\times C_2$ | \( 1 - 3 T + 44 T^{2} - 671 T^{3} + 44 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.ad_bs_azv |
| 71 | $S_4\times C_2$ | \( 1 - 26 T + 401 T^{2} - 3940 T^{3} + 401 p T^{4} - 26 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.aba_pl_afvo |
| 73 | $S_4\times C_2$ | \( 1 - 12 T + 251 T^{2} - 1768 T^{3} + 251 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.am_jr_acqa |
| 79 | $S_4\times C_2$ | \( 1 - 10 T + 255 T^{2} - 1546 T^{3} + 255 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.ak_jv_achm |
| 83 | $S_4\times C_2$ | \( 1 - 5 T + 208 T^{2} - 651 T^{3} + 208 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.af_ia_azb |
| 89 | $S_4\times C_2$ | \( 1 - 9 T + 102 T^{2} - 79 T^{3} + 102 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.aj_dy_adb |
| 97 | $S_4\times C_2$ | \( 1 + 5 T + 114 T^{2} + 1529 T^{3} + 114 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.f_ek_cgv |
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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
−6.88238529165278226843343696578, −6.65886504930547564119806924141, −6.42873126416991343917864316923, −6.40774078656300990271533495691, −6.27181064216357565982553295789, −5.82578643657712762250269158877, −5.55815770643404642171861456018, −5.31099112553104205765048459172, −5.05943717414800359037417777313, −4.88263768077247186926032582544, −4.44712415730881415381015502546, −4.41504090221600070343311286553, −4.14688209570942451571846670305, −3.91698765354196603504692678710, −3.72339074589286317801329685197, −3.47341125456015574886904505345, −3.01877363898489315443522374980, −2.58531132201235076428303161649, −2.46422042111413970829626010786, −2.02199231395715866617695187717, −1.99303347216600128496186578117, −1.23239283104499572428819619652, −0.915674503112207249587972462970, −0.887016352075377794085993163698, −0.56685137283465840171868722860,
0.56685137283465840171868722860, 0.887016352075377794085993163698, 0.915674503112207249587972462970, 1.23239283104499572428819619652, 1.99303347216600128496186578117, 2.02199231395715866617695187717, 2.46422042111413970829626010786, 2.58531132201235076428303161649, 3.01877363898489315443522374980, 3.47341125456015574886904505345, 3.72339074589286317801329685197, 3.91698765354196603504692678710, 4.14688209570942451571846670305, 4.41504090221600070343311286553, 4.44712415730881415381015502546, 4.88263768077247186926032582544, 5.05943717414800359037417777313, 5.31099112553104205765048459172, 5.55815770643404642171861456018, 5.82578643657712762250269158877, 6.27181064216357565982553295789, 6.40774078656300990271533495691, 6.42873126416991343917864316923, 6.65886504930547564119806924141, 6.88238529165278226843343696578
