| L(s) = 1 | − 3·4-s + 4·5-s + 4·16-s − 12·20-s − 16·23-s + 10·25-s − 16·31-s + 12·37-s − 32·47-s − 8·49-s − 12·53-s − 36·59-s − 9·64-s − 36·67-s − 32·71-s + 16·80-s − 4·89-s + 48·92-s + 36·97-s − 30·100-s − 8·103-s − 36·113-s − 64·115-s + 48·124-s + 20·125-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 3/2·4-s + 1.78·5-s + 16-s − 2.68·20-s − 3.33·23-s + 2·25-s − 2.87·31-s + 1.97·37-s − 4.66·47-s − 8/7·49-s − 1.64·53-s − 4.68·59-s − 9/8·64-s − 4.39·67-s − 3.79·71-s + 1.78·80-s − 0.423·89-s + 5.00·92-s + 3.65·97-s − 3·100-s − 0.788·103-s − 3.38·113-s − 5.96·115-s + 4.31·124-s + 1.78·125-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | 3 | | \( 1 \) | |
| 5 | $C_1$ | \( ( 1 - T )^{4} \) | |
| 11 | | \( 1 \) | |
| good | 2 | $C_2^3$ | \( 1 + 3 T^{2} + 5 T^{4} + 3 p^{2} T^{6} + p^{4} T^{8} \) | 4.2.a_d_a_f |
| 7 | $D_4\times C_2$ | \( 1 + 8 T^{2} + 30 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} \) | 4.7.a_i_a_be |
| 13 | $D_4\times C_2$ | \( 1 + 32 T^{2} + 510 T^{4} + 32 p^{2} T^{6} + p^{4} T^{8} \) | 4.13.a_bg_a_tq |
| 17 | $C_2^2$ | \( ( 1 + 31 T^{2} + p^{2} T^{4} )^{2} \) | 4.17.a_ck_a_chf |
| 19 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) | 4.19.a_ca_a_cbu |
| 23 | $C_2^2$ | \( ( 1 + 8 T + 41 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) | 4.23.q_fq_bnk_ikp |
| 29 | $D_4\times C_2$ | \( 1 - 16 T^{2} + 990 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) | 4.29.a_aq_a_bmc |
| 31 | $D_{4}$ | \( ( 1 + 8 T + 57 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) | 4.31.q_gw_cce_nnn |
| 37 | $D_{4}$ | \( ( 1 - 6 T + 62 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.37.am_ge_abts_nrq |
| 41 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) | 4.41.a_fk_a_mfu |
| 43 | $D_4\times C_2$ | \( 1 + 40 T^{2} + 3342 T^{4} + 40 p^{2} T^{6} + p^{4} T^{8} \) | 4.43.a_bo_a_eyo |
| 47 | $D_{4}$ | \( ( 1 + 16 T + 137 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) | 4.47.bg_uk_ism_crxj |
| 53 | $D_{4}$ | \( ( 1 + 6 T + 31 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.53.m_du_bmu_pjv |
| 59 | $D_{4}$ | \( ( 1 + 18 T + 178 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \) | 4.59.bk_bae_mqe_ejsw |
| 61 | $D_4\times C_2$ | \( 1 + 182 T^{2} + 15387 T^{4} + 182 p^{2} T^{6} + p^{4} T^{8} \) | 4.61.a_ha_a_wtv |
| 67 | $D_{4}$ | \( ( 1 + 18 T + 194 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \) | 4.67.bk_bbk_nxk_fdes |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) | 4.71.bg_zs_ncy_fbmc |
| 73 | $D_4\times C_2$ | \( 1 + 20 T^{2} - 1338 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} \) | 4.73.a_u_a_abzm |
| 79 | $D_4\times C_2$ | \( 1 + 278 T^{2} + 31467 T^{4} + 278 p^{2} T^{6} + p^{4} T^{8} \) | 4.79.a_ks_a_buoh |
| 83 | $C_2^2$ | \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \) | 4.83.a_agq_a_bfik |
| 89 | $D_{4}$ | \( ( 1 + 2 T - 10 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.89.e_aq_me_yqo |
| 97 | $D_{4}$ | \( ( 1 - 18 T + 254 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \) | 4.97.abk_bga_assa_iigs |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.05917868455455589890376678266, −5.98565041388651961651421970742, −5.84991854549980199846658892390, −5.54264653117824181508148369327, −5.48963152270294187087984266452, −5.02384960823985081087396886380, −4.85089568771705748818929042326, −4.84853783878581449964537635819, −4.77988828927837230410783425338, −4.31307900552615540390510048110, −4.28717939938248877565057877135, −4.12211849485841805392479364446, −3.97572332703126925086686479006, −3.38784342838606772472197533267, −3.37364134868044061804407374627, −3.15742600485573532902268378871, −3.02942051245569105323832397492, −2.80537086846802370294437404748, −2.53874449056162082812044699124, −1.92113730346377971471638288882, −1.91875412497038827382155470825, −1.78992960535705345976301821544, −1.51069210059155895054811858816, −1.38965919198662787046707628936, −1.15471173239259905997156054136, 0, 0, 0, 0,
1.15471173239259905997156054136, 1.38965919198662787046707628936, 1.51069210059155895054811858816, 1.78992960535705345976301821544, 1.91875412497038827382155470825, 1.92113730346377971471638288882, 2.53874449056162082812044699124, 2.80537086846802370294437404748, 3.02942051245569105323832397492, 3.15742600485573532902268378871, 3.37364134868044061804407374627, 3.38784342838606772472197533267, 3.97572332703126925086686479006, 4.12211849485841805392479364446, 4.28717939938248877565057877135, 4.31307900552615540390510048110, 4.77988828927837230410783425338, 4.84853783878581449964537635819, 4.85089568771705748818929042326, 5.02384960823985081087396886380, 5.48963152270294187087984266452, 5.54264653117824181508148369327, 5.84991854549980199846658892390, 5.98565041388651961651421970742, 6.05917868455455589890376678266
