| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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.d_m_bn |
| 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.ad_y_acl |
| 13 | $A_4\times C_2$ | \( 1 + 36 T^{2} - T^{3} + 36 p T^{4} + p^{3} T^{6} \) | 3.13.a_bk_ab |
| 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.g_ci_ir |
| 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.m_ds_xz |
| 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.ag_cx_alg |
| 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.am_fc_abhp |
| 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.g_eq_tf |
| 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.g_ek_pp |
| 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.am_ds_avx |
| 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.p_cu_an |
| 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_bb |
| 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_aev |
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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
−8.311948651069349744819767551472, −7.73028706925573393867744987864, −7.64081839791476466401844205292, −7.39168005416505466723638625154, −7.37303248041090200257910981749, −6.82391171862215724031173377921, −6.57428916355476740976062191620, −6.03918749804845099822220476770, −6.00176020919290467996082861047, −5.94684838449887350896432404839, −5.81239313682952076637644364775, −5.54012307722956620691961158818, −5.16875187628295276348036339993, −4.44649816353458934374621721637, −4.37130543938252912023377598279, −4.33595985182479831217567862412, −3.71554757525713149617666034404, −3.48420575237036852363408608569, −3.48266684265518241467012949813, −2.71399180991414217572485245461, −2.70689353041491739251184309421, −2.54567399641625037791538540884, −1.70553176829234427971352495513, −1.70503818560569579679607702043, −1.18792361312193777263720358567, 0, 0, 0,
1.18792361312193777263720358567, 1.70503818560569579679607702043, 1.70553176829234427971352495513, 2.54567399641625037791538540884, 2.70689353041491739251184309421, 2.71399180991414217572485245461, 3.48266684265518241467012949813, 3.48420575237036852363408608569, 3.71554757525713149617666034404, 4.33595985182479831217567862412, 4.37130543938252912023377598279, 4.44649816353458934374621721637, 5.16875187628295276348036339993, 5.54012307722956620691961158818, 5.81239313682952076637644364775, 5.94684838449887350896432404839, 6.00176020919290467996082861047, 6.03918749804845099822220476770, 6.57428916355476740976062191620, 6.82391171862215724031173377921, 7.37303248041090200257910981749, 7.39168005416505466723638625154, 7.64081839791476466401844205292, 7.73028706925573393867744987864, 8.311948651069349744819767551472
