| L(s) = 1 | − 2·4-s − 4·5-s + 16-s + 8·20-s + 10·25-s − 8·31-s − 8·37-s − 16·49-s − 24·53-s − 24·59-s − 4·64-s + 16·67-s − 24·71-s − 4·80-s + 24·89-s + 8·97-s − 20·100-s + 32·103-s − 24·113-s + 16·124-s − 20·125-s + 127-s + 131-s + 137-s + 139-s + 16·148-s + 149-s + ⋯ |
| L(s) = 1 | − 4-s − 1.78·5-s + 1/4·16-s + 1.78·20-s + 2·25-s − 1.43·31-s − 1.31·37-s − 2.28·49-s − 3.29·53-s − 3.12·59-s − 1/2·64-s + 1.95·67-s − 2.84·71-s − 0.447·80-s + 2.54·89-s + 0.812·97-s − 2·100-s + 3.15·103-s − 2.25·113-s + 1.43·124-s − 1.78·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.31·148-s + 0.0819·149-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 | $C_2^2 \wr C_2$ | \( 1 + p T^{2} + 3 T^{4} + p^{3} T^{6} + p^{4} T^{8} \) | 4.2.a_c_a_d |
| 7 | $C_2^2 \wr C_2$ | \( 1 + 16 T^{2} + 138 T^{4} + 16 p^{2} T^{6} + p^{4} T^{8} \) | 4.7.a_q_a_fi |
| 13 | $C_2^2 \wr C_2$ | \( 1 + 16 T^{2} + 186 T^{4} + 16 p^{2} T^{6} + p^{4} T^{8} \) | 4.13.a_q_a_he |
| 17 | $C_2^2 \wr C_2$ | \( 1 + 56 T^{2} + 1338 T^{4} + 56 p^{2} T^{6} + p^{4} T^{8} \) | 4.17.a_ce_a_bzm |
| 19 | $C_2^2 \wr C_2$ | \( 1 + 52 T^{2} + 1302 T^{4} + 52 p^{2} T^{6} + p^{4} T^{8} \) | 4.19.a_ca_a_byc |
| 23 | $C_2^2$ | \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \) | 4.23.a_bs_a_chi |
| 29 | $C_2^2 \wr C_2$ | \( 1 + 44 T^{2} + 1302 T^{4} + 44 p^{2} T^{6} + p^{4} T^{8} \) | 4.29.a_bs_a_byc |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) | 4.31.i_fs_bdw_ktq |
| 37 | $D_{4}$ | \( ( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.37.i_eu_bca_kda |
| 41 | $C_2^2 \wr C_2$ | \( 1 + 92 T^{2} + 5382 T^{4} + 92 p^{2} T^{6} + p^{4} T^{8} \) | 4.41.a_do_a_hza |
| 43 | $C_2^2 \wr C_2$ | \( 1 + 160 T^{2} + 10074 T^{4} + 160 p^{2} T^{6} + p^{4} T^{8} \) | 4.43.a_ge_a_oxm |
| 47 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) | 4.47.a_ae_a_goc |
| 53 | $D_{4}$ | \( ( 1 + 12 T + 118 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) | 4.53.y_oq_gbw_bzms |
| 59 | $D_{4}$ | \( ( 1 + 12 T + 130 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) | 4.59.y_po_gsm_cili |
| 61 | $C_2^2 \wr C_2$ | \( 1 + 100 T^{2} + 9558 T^{4} + 100 p^{2} T^{6} + p^{4} T^{8} \) | 4.61.a_dw_a_odq |
| 67 | $D_{4}$ | \( ( 1 - 8 T + 126 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) | 4.67.aq_me_aeou_bxlu |
| 71 | $D_{4}$ | \( ( 1 + 12 T + 154 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) | 4.71.y_rk_hzs_dcgk |
| 73 | $C_2^2 \wr C_2$ | \( 1 + 160 T^{2} + 13002 T^{4} + 160 p^{2} T^{6} + p^{4} T^{8} \) | 4.73.a_ge_a_tgc |
| 79 | $C_2^2 \wr C_2$ | \( 1 - 92 T^{2} + 2982 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} \) | 4.79.a_ado_a_eks |
| 83 | $C_2^2 \wr C_2$ | \( 1 + 296 T^{2} + 35658 T^{4} + 296 p^{2} T^{6} + p^{4} T^{8} \) | 4.83.a_lk_a_catm |
| 89 | $D_{4}$ | \( ( 1 - 12 T + 118 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) | 4.89.ay_oq_ahjc_ddys |
| 97 | $D_{4}$ | \( ( 1 - 4 T - 18 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.97.ai_au_ayi_bgxq |
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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
−6.20796710599121307631541529954, −6.05918369383212241082345996136, −5.61705980175067973614507497083, −5.46776143525984803981437509454, −5.34518470829807638272800513989, −4.99045895173595680234393165349, −4.88073689489393409616362264480, −4.80224827357890861900771394492, −4.65771392060218034307001365634, −4.35994250101531844519013112201, −4.26383394365143085023195128156, −4.07790753065333851054698253189, −3.75649584370811358765198395695, −3.46143795523770631537111800186, −3.37080903578311124597048809913, −3.27396086203878247747772280890, −3.21726047307933092892602328440, −2.88606703393221000419150491996, −2.55800243756669542943504213092, −2.10855091449907648832633109926, −1.92345488943174495927667102536, −1.90382855195051169321338648264, −1.21592136448432246898512883653, −1.12733855695365154973184879866, −1.12547456000065746731045234235, 0, 0, 0, 0,
1.12547456000065746731045234235, 1.12733855695365154973184879866, 1.21592136448432246898512883653, 1.90382855195051169321338648264, 1.92345488943174495927667102536, 2.10855091449907648832633109926, 2.55800243756669542943504213092, 2.88606703393221000419150491996, 3.21726047307933092892602328440, 3.27396086203878247747772280890, 3.37080903578311124597048809913, 3.46143795523770631537111800186, 3.75649584370811358765198395695, 4.07790753065333851054698253189, 4.26383394365143085023195128156, 4.35994250101531844519013112201, 4.65771392060218034307001365634, 4.80224827357890861900771394492, 4.88073689489393409616362264480, 4.99045895173595680234393165349, 5.34518470829807638272800513989, 5.46776143525984803981437509454, 5.61705980175067973614507497083, 6.05918369383212241082345996136, 6.20796710599121307631541529954
