| L(s) = 1 | + 3·2-s + 6·4-s + 2·5-s + 4·7-s + 10·8-s + 6·10-s + 3·11-s − 6·13-s + 12·14-s + 15·16-s + 2·17-s − 3·19-s + 12·20-s + 9·22-s − 3·25-s − 18·26-s + 24·28-s + 10·29-s + 21·32-s + 6·34-s + 8·35-s + 14·37-s − 9·38-s + 20·40-s + 14·41-s + 20·43-s + 18·44-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 3·4-s + 0.894·5-s + 1.51·7-s + 3.53·8-s + 1.89·10-s + 0.904·11-s − 1.66·13-s + 3.20·14-s + 15/4·16-s + 0.485·17-s − 0.688·19-s + 2.68·20-s + 1.91·22-s − 3/5·25-s − 3.53·26-s + 4.53·28-s + 1.85·29-s + 3.71·32-s + 1.02·34-s + 1.35·35-s + 2.30·37-s − 1.45·38-s + 3.16·40-s + 2.18·41-s + 3.04·43-s + 2.71·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{6} \cdot 11^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{6} \cdot 11^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(40.41573818\) |
| \(L(\frac12)\) |
\(\approx\) |
\(40.41573818\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{3} \) | |
| 3 | | \( 1 \) | |
| 11 | $C_1$ | \( ( 1 - T )^{3} \) | |
| 19 | $C_1$ | \( ( 1 + T )^{3} \) | |
| good | 5 | $A_4\times C_2$ | \( 1 - 2 T + 7 T^{2} - 12 T^{3} + 7 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.5.ac_h_am |
| 7 | $A_4\times C_2$ | \( 1 - 4 T + 17 T^{2} - 48 T^{3} + 17 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.7.ae_r_abw |
| 13 | $A_4\times C_2$ | \( 1 + 6 T + 23 T^{2} + 4 p T^{3} + 23 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.g_x_ca |
| 17 | $A_4\times C_2$ | \( 1 - 2 T + 15 T^{2} - 60 T^{3} + 15 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.ac_p_aci |
| 23 | $A_4\times C_2$ | \( 1 + 41 T^{2} + 56 T^{3} + 41 p T^{4} + p^{3} T^{6} \) | 3.23.a_bp_ce |
| 29 | $A_4\times C_2$ | \( 1 - 10 T + 83 T^{2} - 476 T^{3} + 83 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.ak_df_asi |
| 31 | $A_4\times C_2$ | \( 1 + 9 T^{2} - 56 T^{3} + 9 p T^{4} + p^{3} T^{6} \) | 3.31.a_j_ace |
| 37 | $A_4\times C_2$ | \( 1 - 14 T + 167 T^{2} - 1092 T^{3} + 167 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.ao_gl_abqa |
| 41 | $A_4\times C_2$ | \( 1 - 14 T + 151 T^{2} - 1092 T^{3} + 151 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.ao_fv_abqa |
| 43 | $C_6$ | \( 1 - 20 T + 225 T^{2} - 1784 T^{3} + 225 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.au_ir_acqq |
| 47 | $A_4\times C_2$ | \( 1 - 8 T - 15 T^{2} + 600 T^{3} - 15 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.ai_ap_xc |
| 53 | $A_4\times C_2$ | \( 1 - 18 T + 239 T^{2} - 1900 T^{3} + 239 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.as_jf_acvc |
| 59 | $A_4\times C_2$ | \( 1 - 8 T + 161 T^{2} - 880 T^{3} + 161 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.ai_gf_abhw |
| 61 | $A_4\times C_2$ | \( 1 + 14 T + 3 p T^{2} + 1652 T^{3} + 3 p^{2} T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.o_hb_clo |
| 67 | $A_4\times C_2$ | \( 1 + 8 T + 73 T^{2} + 560 T^{3} + 73 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.i_cv_vo |
| 71 | $A_4\times C_2$ | \( 1 - 20 T + 281 T^{2} - 2512 T^{3} + 281 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.au_kv_adsq |
| 73 | $A_4\times C_2$ | \( 1 + 6 T + 119 T^{2} + 1108 T^{3} + 119 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.g_ep_bqq |
| 79 | $A_4\times C_2$ | \( 1 + 8 T + 249 T^{2} + 1256 T^{3} + 249 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.i_jp_bwi |
| 83 | $A_4\times C_2$ | \( 1 + 8 T + 121 T^{2} + 816 T^{3} + 121 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.i_er_bfk |
| 89 | $A_4\times C_2$ | \( 1 - 22 T + 391 T^{2} - 4020 T^{3} + 391 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.aw_pb_afyq |
| 97 | $A_4\times C_2$ | \( 1 + 22 T + 415 T^{2} + 4372 T^{3} + 415 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.w_pz_gme |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.44682054958944737028279607195, −7.17860613277960991119814467772, −7.10994296899600796270167935869, −6.69893354817945667751844635082, −6.23182090385327740893514708460, −6.16602282950869711130542662884, −5.96192135490596879967943867638, −5.76215645586787104653238743571, −5.53682026295812514250822471022, −5.28840617772501529664980107599, −4.72576767566446663304417166976, −4.67958244700123106472178394899, −4.54619888734303240117600527482, −4.31684759701775062910909674808, −3.97173538698417761786861572518, −3.88781100965638396008475215460, −3.21297153004119890590030797705, −3.05839846974890370877863264835, −2.55755315982937305210338581339, −2.23885777350957250045118136016, −2.19670041353341317534415050854, −2.19053230056785224681873427254, −1.30244102196952541630099255480, −0.989166736159533695213484769145, −0.831719516099337876024932480099,
0.831719516099337876024932480099, 0.989166736159533695213484769145, 1.30244102196952541630099255480, 2.19053230056785224681873427254, 2.19670041353341317534415050854, 2.23885777350957250045118136016, 2.55755315982937305210338581339, 3.05839846974890370877863264835, 3.21297153004119890590030797705, 3.88781100965638396008475215460, 3.97173538698417761786861572518, 4.31684759701775062910909674808, 4.54619888734303240117600527482, 4.67958244700123106472178394899, 4.72576767566446663304417166976, 5.28840617772501529664980107599, 5.53682026295812514250822471022, 5.76215645586787104653238743571, 5.96192135490596879967943867638, 6.16602282950869711130542662884, 6.23182090385327740893514708460, 6.69893354817945667751844635082, 7.10994296899600796270167935869, 7.17860613277960991119814467772, 7.44682054958944737028279607195
