| L(s) = 1 | + 2-s − 3·3-s − 2·4-s + 4·5-s − 3·6-s − 3·7-s − 2·8-s + 6·9-s + 4·10-s + 6·12-s + 6·13-s − 3·14-s − 12·15-s + 16-s + 8·17-s + 6·18-s + 3·19-s − 8·20-s + 9·21-s − 4·23-s + 6·24-s + 25-s + 6·26-s − 10·27-s + 6·28-s + 20·29-s − 12·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.73·3-s − 4-s + 1.78·5-s − 1.22·6-s − 1.13·7-s − 0.707·8-s + 2·9-s + 1.26·10-s + 1.73·12-s + 1.66·13-s − 0.801·14-s − 3.09·15-s + 1/4·16-s + 1.94·17-s + 1.41·18-s + 0.688·19-s − 1.78·20-s + 1.96·21-s − 0.834·23-s + 1.22·24-s + 1/5·25-s + 1.17·26-s − 1.92·27-s + 1.13·28-s + 3.71·29-s − 2.19·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63521199 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63521199 ^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.858514631\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.858514631\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{3} \) | |
| 7 | $C_1$ | \( ( 1 + T )^{3} \) | |
| 19 | $C_1$ | \( ( 1 - T )^{3} \) | |
| good | 2 | $S_4\times C_2$ | \( 1 - T + 3 T^{2} - 3 T^{3} + 3 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) | 3.2.ab_d_ad |
| 5 | $S_4\times C_2$ | \( 1 - 4 T + 3 p T^{2} - 36 T^{3} + 3 p^{2} T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.5.ae_p_abk |
| 11 | $S_4\times C_2$ | \( 1 + 17 T^{2} - 16 T^{3} + 17 p T^{4} + p^{3} T^{6} \) | 3.11.a_r_aq |
| 13 | $S_4\times C_2$ | \( 1 - 6 T + 35 T^{2} - 148 T^{3} + 35 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.ag_bj_afs |
| 17 | $S_4\times C_2$ | \( 1 - 8 T + 67 T^{2} - 276 T^{3} + 67 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.ai_cp_akq |
| 23 | $S_4\times C_2$ | \( 1 + 4 T - 11 T^{2} - 216 T^{3} - 11 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.e_al_aii |
| 29 | $S_4\times C_2$ | \( 1 - 20 T + 211 T^{2} - 1404 T^{3} + 211 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.au_id_acca |
| 31 | $S_4\times C_2$ | \( 1 - 8 T + 53 T^{2} - 192 T^{3} + 53 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.ai_cb_ahk |
| 37 | $S_4\times C_2$ | \( 1 + 10 T + 91 T^{2} + 604 T^{3} + 91 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.k_dn_xg |
| 41 | $S_4\times C_2$ | \( 1 - 14 T + 135 T^{2} - 852 T^{3} + 135 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.ao_ff_abgu |
| 43 | $S_4\times C_2$ | \( 1 + 8 T + 89 T^{2} + 384 T^{3} + 89 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.i_dl_ou |
| 47 | $S_4\times C_2$ | \( 1 - 6 T + 53 T^{2} - 640 T^{3} + 53 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.ag_cb_ayq |
| 53 | $S_4\times C_2$ | \( 1 - 4 T + 131 T^{2} - 308 T^{3} + 131 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.ae_fb_alw |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{3} \) | 3.59.am_ir_acey |
| 61 | $S_4\times C_2$ | \( 1 - 14 T + 195 T^{2} - 1556 T^{3} + 195 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.ao_hn_achw |
| 67 | $S_4\times C_2$ | \( 1 + 4 T + 169 T^{2} + 568 T^{3} + 169 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.e_gn_vw |
| 71 | $S_4\times C_2$ | \( 1 - 2 T + 121 T^{2} - 552 T^{3} + 121 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.ac_er_avg |
| 73 | $S_4\times C_2$ | \( 1 + 10 T + 231 T^{2} + 1420 T^{3} + 231 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.k_ix_ccq |
| 79 | $S_4\times C_2$ | \( 1 + 16 T + 173 T^{2} + 1248 T^{3} + 173 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.q_gr_bwa |
| 83 | $S_4\times C_2$ | \( 1 + 14 T + 161 T^{2} + 1360 T^{3} + 161 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.o_gf_cai |
| 89 | $S_4\times C_2$ | \( 1 - 14 T + 279 T^{2} - 2196 T^{3} + 279 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.ao_kt_adgm |
| 97 | $S_4\times C_2$ | \( 1 - 10 T + 111 T^{2} - 76 T^{3} + 111 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.ak_eh_acy |
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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
−10.13204044581874548850357935595, −9.939662594440579997386953595328, −9.701832327051288292767229483536, −9.424381366107457693598515701595, −8.647000241558964330614010197193, −8.609194115024463328230152337136, −8.560184680708139936944214663155, −7.67439858523015733268419373339, −7.58671483002173718105520692937, −6.80201813165663562901847856350, −6.74674450402539244869251036375, −6.19288926420438796252546173097, −6.19202194877736968956589208034, −5.65955622542699280757513678320, −5.51105934802307178463741765682, −5.49323216642143360717978901928, −4.65826845090344499188971492955, −4.61476875792048678066000503729, −4.06164130812424243500959980106, −3.69038859227705020247129944534, −3.21591763127668172356756531832, −2.72807552975338520682386566388, −1.96711390167257975936627033293, −1.04683135690396239490847441881, −0.926377723373993413653802739401,
0.926377723373993413653802739401, 1.04683135690396239490847441881, 1.96711390167257975936627033293, 2.72807552975338520682386566388, 3.21591763127668172356756531832, 3.69038859227705020247129944534, 4.06164130812424243500959980106, 4.61476875792048678066000503729, 4.65826845090344499188971492955, 5.49323216642143360717978901928, 5.51105934802307178463741765682, 5.65955622542699280757513678320, 6.19202194877736968956589208034, 6.19288926420438796252546173097, 6.74674450402539244869251036375, 6.80201813165663562901847856350, 7.58671483002173718105520692937, 7.67439858523015733268419373339, 8.560184680708139936944214663155, 8.609194115024463328230152337136, 8.647000241558964330614010197193, 9.424381366107457693598515701595, 9.701832327051288292767229483536, 9.939662594440579997386953595328, 10.13204044581874548850357935595
