| L(s) = 1 | + 4·7-s + 4·13-s − 5·16-s + 24·19-s + 4·31-s + 24·37-s + 6·49-s − 24·61-s + 20·67-s + 4·73-s − 4·79-s + 16·91-s − 4·97-s − 4·103-s + 28·109-s − 20·112-s + 127-s + 131-s + 96·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 18·169-s + ⋯ |
| L(s) = 1 | + 1.51·7-s + 1.10·13-s − 5/4·16-s + 5.50·19-s + 0.718·31-s + 3.94·37-s + 6/7·49-s − 3.07·61-s + 2.44·67-s + 0.468·73-s − 0.450·79-s + 1.67·91-s − 0.406·97-s − 0.394·103-s + 2.68·109-s − 1.88·112-s + 0.0887·127-s + 0.0873·131-s + 8.32·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.38·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(13.55558492\) |
| \(L(\frac12)\) |
\(\approx\) |
\(13.55558492\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 11 | | \( 1 \) | |
| good | 2 | $D_4\times C_2$ | \( 1 + 5 T^{4} + p^{4} T^{8} \) | 4.2.a_a_a_f |
| 5 | $D_4\times C_2$ | \( 1 + 2 T^{4} + p^{4} T^{8} \) | 4.5.a_a_a_c |
| 7 | $D_{4}$ | \( ( 1 - 2 T + 3 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.7.ae_k_abo_gh |
| 13 | $D_{4}$ | \( ( 1 - 2 T + 15 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.13.ae_bi_aei_zr |
| 17 | $C_2^2 \wr C_2$ | \( 1 + 24 T^{2} + 290 T^{4} + 24 p^{2} T^{6} + p^{4} T^{8} \) | 4.17.a_y_a_le |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) | 4.19.ay_lg_adhw_rgw |
| 23 | $C_2^2 \wr C_2$ | \( 1 + 48 T^{2} + 1202 T^{4} + 48 p^{2} T^{6} + p^{4} T^{8} \) | 4.23.a_bw_a_bug |
| 29 | $C_2^2 \wr C_2$ | \( 1 + 20 T^{2} + 1350 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} \) | 4.29.a_u_a_bzy |
| 31 | $D_{4}$ | \( ( 1 - 2 T + 51 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.31.ae_ec_amq_hbn |
| 37 | $D_{4}$ | \( ( 1 - 12 T + 98 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) | 4.37.ay_nc_aeuq_biao |
| 41 | $C_2^2 \wr C_2$ | \( 1 + 144 T^{2} + 8498 T^{4} + 144 p^{2} T^{6} + p^{4} T^{8} \) | 4.41.a_fo_a_mow |
| 43 | $C_2^2$ | \( ( 1 + 74 T^{2} + p^{2} T^{4} )^{2} \) | 4.43.a_fs_a_now |
| 47 | $C_2^2 \wr C_2$ | \( 1 + 24 T^{2} + 4130 T^{4} + 24 p^{2} T^{6} + p^{4} T^{8} \) | 4.47.a_y_a_gcw |
| 53 | $C_2^2 \wr C_2$ | \( 1 - 28 T^{2} - 1098 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} \) | 4.53.a_abc_a_abqg |
| 59 | $C_2^2 \wr C_2$ | \( 1 + 72 T^{2} + 7826 T^{4} + 72 p^{2} T^{6} + p^{4} T^{8} \) | 4.59.a_cu_a_lpa |
| 61 | $D_{4}$ | \( ( 1 + 12 T + 146 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) | 4.61.y_qu_hjc_cqnu |
| 67 | $D_{4}$ | \( ( 1 - 10 T + 147 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) | 4.67.au_pe_agiq_cnbv |
| 71 | $C_2^2 \wr C_2$ | \( 1 + 188 T^{2} + 18486 T^{4} + 188 p^{2} T^{6} + p^{4} T^{8} \) | 4.71.a_hg_a_bbja |
| 73 | $D_{4}$ | \( ( 1 - 2 T + 39 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.73.ae_de_arg_swx |
| 79 | $D_{4}$ | \( ( 1 + 2 T + 111 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.79.e_is_bdg_blqh |
| 83 | $C_2^2 \wr C_2$ | \( 1 + 44 T^{2} + 10374 T^{4} + 44 p^{2} T^{6} + p^{4} T^{8} \) | 4.83.a_bs_a_pja |
| 89 | $C_2^2 \wr C_2$ | \( 1 + 260 T^{2} + 32310 T^{4} + 260 p^{2} T^{6} + p^{4} T^{8} \) | 4.89.a_ka_a_bvus |
| 97 | $D_{4}$ | \( ( 1 + 2 T + 87 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.97.e_gw_bci_boet |
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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
−6.13041257473052440305239636202, −5.88136615558264038784196653274, −5.66328153176260529496380472635, −5.54742216181561404326565719920, −5.33262531851854506760174023755, −4.98849935417913726636873268028, −4.93330676994928867441432771551, −4.69309202380570439591968309178, −4.67752188894814759368605706401, −4.29357851499915204024534887329, −4.15396360618518819362603447977, −3.75287776562278724758341676954, −3.62182196215707250471438849148, −3.46993799081013434308403417133, −3.01890921746735937383754836996, −2.95346684571583527207763773638, −2.67996391516909008473124772646, −2.62688203190604691318450513849, −2.18771401003319877001840809780, −1.73210194668991165863075802694, −1.58765758135422894900010331651, −1.29766023324857602157850878008, −1.03244407174170049809091159895, −0.71409037177593162071487009011, −0.64881010755211239835673303932,
0.64881010755211239835673303932, 0.71409037177593162071487009011, 1.03244407174170049809091159895, 1.29766023324857602157850878008, 1.58765758135422894900010331651, 1.73210194668991165863075802694, 2.18771401003319877001840809780, 2.62688203190604691318450513849, 2.67996391516909008473124772646, 2.95346684571583527207763773638, 3.01890921746735937383754836996, 3.46993799081013434308403417133, 3.62182196215707250471438849148, 3.75287776562278724758341676954, 4.15396360618518819362603447977, 4.29357851499915204024534887329, 4.67752188894814759368605706401, 4.69309202380570439591968309178, 4.93330676994928867441432771551, 4.98849935417913726636873268028, 5.33262531851854506760174023755, 5.54742216181561404326565719920, 5.66328153176260529496380472635, 5.88136615558264038784196653274, 6.13041257473052440305239636202
