| L(s) = 1 | + 3·7-s − 6·9-s − 3·11-s + 6·23-s − 12·25-s − 27-s + 12·29-s + 3·31-s − 6·37-s − 12·41-s + 6·43-s + 6·47-s − 3·49-s − 12·53-s + 6·59-s − 12·61-s − 18·63-s − 18·67-s − 18·71-s − 24·73-s − 9·77-s − 24·79-s + 18·81-s + 15·83-s + 18·99-s − 12·101-s − 3·103-s + ⋯ |
| L(s) = 1 | + 1.13·7-s − 2·9-s − 0.904·11-s + 1.25·23-s − 2.39·25-s − 0.192·27-s + 2.22·29-s + 0.538·31-s − 0.986·37-s − 1.87·41-s + 0.914·43-s + 0.875·47-s − 3/7·49-s − 1.64·53-s + 0.781·59-s − 1.53·61-s − 2.26·63-s − 2.19·67-s − 2.13·71-s − 2.80·73-s − 1.02·77-s − 2.70·79-s + 2·81-s + 1.64·83-s + 1.80·99-s − 1.19·101-s − 0.295·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | 2 | | \( 1 \) | |
| 19 | | \( 1 \) | |
| good | 3 | $A_4\times C_2$ | \( 1 + 2 p T^{2} + T^{3} + 2 p^{2} T^{4} + p^{3} T^{6} \) | 3.3.a_g_b |
| 5 | $A_4\times C_2$ | \( 1 + 12 T^{2} + T^{3} + 12 p T^{4} + p^{3} T^{6} \) | 3.5.a_m_b |
| 7 | $A_4\times C_2$ | \( 1 - 3 T + 12 T^{2} - 39 T^{3} + 12 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.7.ad_m_abn |
| 11 | $A_4\times C_2$ | \( 1 + 3 T + 24 T^{2} + 63 T^{3} + 24 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.d_y_cl |
| 13 | $A_4\times C_2$ | \( 1 + 36 T^{2} + T^{3} + 36 p T^{4} + p^{3} T^{6} \) | 3.13.a_bk_b |
| 17 | $A_4\times C_2$ | \( 1 + 48 T^{2} + T^{3} + 48 p T^{4} + p^{3} T^{6} \) | 3.17.a_bw_b |
| 23 | $A_4\times C_2$ | \( 1 - 6 T + 60 T^{2} - 225 T^{3} + 60 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.ag_ci_air |
| 29 | $A_4\times C_2$ | \( 1 - 12 T + 96 T^{2} - 623 T^{3} + 96 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.am_ds_axz |
| 31 | $A_4\times C_2$ | \( 1 - 3 T + 60 T^{2} - 79 T^{3} + 60 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.ad_ci_adb |
| 37 | $A_4\times C_2$ | \( 1 + 6 T + 75 T^{2} + 292 T^{3} + 75 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.g_cx_lg |
| 41 | $A_4\times C_2$ | \( 1 + 12 T + 132 T^{2} + 873 T^{3} + 132 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.m_fc_bhp |
| 43 | $A_4\times C_2$ | \( 1 - 6 T + 120 T^{2} - 499 T^{3} + 120 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.ag_eq_atf |
| 47 | $A_4\times C_2$ | \( 1 - 6 T + 114 T^{2} - 405 T^{3} + 114 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.ag_ek_app |
| 53 | $A_4\times C_2$ | \( 1 + 12 T + 96 T^{2} + 569 T^{3} + 96 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.m_ds_vx |
| 59 | $A_4\times C_2$ | \( 1 - 6 T + 114 T^{2} - 691 T^{3} + 114 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.ag_ek_abap |
| 61 | $A_4\times C_2$ | \( 1 + 12 T + 84 T^{2} + 257 T^{3} + 84 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.m_dg_jx |
| 67 | $A_4\times C_2$ | \( 1 + 18 T + 282 T^{2} + 2439 T^{3} + 282 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.s_kw_dpv |
| 71 | $A_4\times C_2$ | \( 1 + 18 T + 204 T^{2} + 1593 T^{3} + 204 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.s_hw_cjh |
| 73 | $A_4\times C_2$ | \( 1 + 24 T + 264 T^{2} + 2157 T^{3} + 264 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.y_ke_dez |
| 79 | $A_4\times C_2$ | \( 1 + 24 T + 402 T^{2} + 4061 T^{3} + 402 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.y_pm_gaf |
| 83 | $A_4\times C_2$ | \( 1 - 15 T + 72 T^{2} + 13 T^{3} + 72 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.ap_cu_n |
| 89 | $A_4\times C_2$ | \( 1 + 240 T^{2} - 27 T^{3} + 240 p T^{4} + p^{3} T^{6} \) | 3.89.a_jg_abb |
| 97 | $A_4\times C_2$ | \( 1 + 216 T^{2} + 125 T^{3} + 216 p T^{4} + p^{3} T^{6} \) | 3.97.a_ii_ev |
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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
−7.67860348324222142053989157665, −7.21251198257970835084985487540, −7.11967038408695502082499020621, −7.00698642066327844377397813168, −6.39339795897402208562985490465, −6.28007165953469765856533768395, −6.08602857203466359731765805539, −5.79387209733570166221911874464, −5.59241897489915924543024279731, −5.46898494989016865819234499716, −5.02860471725911352926052044851, −4.93228219239056951632296059735, −4.57651421114027559774947975046, −4.52478640193491950395367890722, −4.01456529952219604031198054256, −3.97230650458664271875486663547, −3.16447710988846746421659310281, −3.14497531647791671477894791960, −3.11661045025212687472974727959, −2.67509158632927415108739715817, −2.37188052866915527639142499354, −2.19416765775757011133889567482, −1.48268301837234460855463757775, −1.34736458719037820257427915226, −1.25224911698167647637461150590, 0, 0, 0,
1.25224911698167647637461150590, 1.34736458719037820257427915226, 1.48268301837234460855463757775, 2.19416765775757011133889567482, 2.37188052866915527639142499354, 2.67509158632927415108739715817, 3.11661045025212687472974727959, 3.14497531647791671477894791960, 3.16447710988846746421659310281, 3.97230650458664271875486663547, 4.01456529952219604031198054256, 4.52478640193491950395367890722, 4.57651421114027559774947975046, 4.93228219239056951632296059735, 5.02860471725911352926052044851, 5.46898494989016865819234499716, 5.59241897489915924543024279731, 5.79387209733570166221911874464, 6.08602857203466359731765805539, 6.28007165953469765856533768395, 6.39339795897402208562985490465, 7.00698642066327844377397813168, 7.11967038408695502082499020621, 7.21251198257970835084985487540, 7.67860348324222142053989157665
