| L(s) = 1 | + 2·2-s + 2·4-s + 4·7-s + 4·8-s + 4·9-s + 8·14-s + 8·16-s + 8·18-s + 4·23-s + 8·28-s − 8·31-s + 8·32-s + 8·36-s − 8·41-s + 8·46-s − 20·47-s − 12·49-s + 16·56-s − 16·62-s + 16·63-s + 8·64-s + 8·71-s + 16·72-s − 16·73-s − 32·79-s + 6·81-s − 16·82-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 1.51·7-s + 1.41·8-s + 4/3·9-s + 2.13·14-s + 2·16-s + 1.88·18-s + 0.834·23-s + 1.51·28-s − 1.43·31-s + 1.41·32-s + 4/3·36-s − 1.24·41-s + 1.17·46-s − 2.91·47-s − 1.71·49-s + 2.13·56-s − 2.03·62-s + 2.01·63-s + 64-s + 0.949·71-s + 1.88·72-s − 1.87·73-s − 3.60·79-s + 2/3·81-s − 1.76·82-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.875858435\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.875858435\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} \) | |
| 5 | | \( 1 \) | |
| good | 3 | $D_4\times C_2$ | \( 1 - 4 T^{2} + 10 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) | 4.3.a_ae_a_k |
| 7 | $D_{4}$ | \( ( 1 - 2 T + 12 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.7.ae_bc_acy_lm |
| 11 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) | 4.11.a_abk_a_vu |
| 13 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) | 4.13.a_abc_a_uo |
| 17 | $C_2^2$ | \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \) | 4.17.a_bs_a_bow |
| 19 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 54 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) | 4.19.a_au_a_cc |
| 23 | $D_{4}$ | \( ( 1 - 2 T + 20 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.23.ae_bs_agq_cle |
| 29 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) | 4.29.a_au_a_cqo |
| 31 | $D_{4}$ | \( ( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.31.i_eu_bae_iqg |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) | 4.37.a_afk_a_lhu |
| 41 | $D_{4}$ | \( ( 1 + 4 T + 74 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.41.i_gi_bjk_pak |
| 43 | $D_4\times C_2$ | \( 1 - 68 T^{2} + 4266 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \) | 4.43.a_acq_a_gic |
| 47 | $D_{4}$ | \( ( 1 + 10 T + 116 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) | 4.47.u_mu_evk_boja |
| 53 | $D_4\times C_2$ | \( 1 - 60 T^{2} + 3446 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} \) | 4.53.a_aci_a_fco |
| 59 | $D_4\times C_2$ | \( 1 - 180 T^{2} + 14294 T^{4} - 180 p^{2} T^{6} + p^{4} T^{8} \) | 4.59.a_agy_a_vdu |
| 61 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 11574 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) | 4.61.a_afk_a_rde |
| 67 | $D_4\times C_2$ | \( 1 - 100 T^{2} + 10506 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \) | 4.67.a_adw_a_poc |
| 71 | $D_{4}$ | \( ( 1 - 4 T + 134 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.71.ai_ky_aclc_bsvu |
| 73 | $D_{4}$ | \( ( 1 + 8 T + 150 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) | 4.73.q_oa_fhg_ckws |
| 79 | $D_{4}$ | \( ( 1 + 16 T + 174 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) | 4.79.bg_xg_lzk_etcg |
| 83 | $D_4\times C_2$ | \( 1 - 308 T^{2} + 37386 T^{4} - 308 p^{2} T^{6} + p^{4} T^{8} \) | 4.83.a_alw_a_cdhy |
| 89 | $D_{4}$ | \( ( 1 - 4 T + 134 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.89.ai_ky_acqq_ccfm |
| 97 | $D_{4}$ | \( ( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) | 4.97.aq_ki_aesm_cjpm |
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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
−9.013758836363201580201155147901, −8.972932572654809180346407228696, −8.388878055674106088832302406700, −8.237881899898414674359498437509, −7.926355947701997810716764479918, −7.86157264842483511298107402869, −7.34284845134376802528979596671, −7.28601238535297850720143320124, −6.83291220485538776692749363331, −6.74762410876947136090532186280, −6.42725474082146945134677209836, −5.85041551936445313113891560737, −5.70941848086089266445339497339, −5.18600007951943416347800487239, −5.02882779130563354887572436271, −4.87015517522962115371896364688, −4.50891841420106988357796324547, −4.29972863374502112967357765166, −3.97055997904480693761724235852, −3.39480704296945405252045678404, −3.27991297599157474048785793937, −2.78268952173905654153624522446, −1.83342274361543785051074145781, −1.60987450182671263655638784436, −1.53090805213268930119725659074,
1.53090805213268930119725659074, 1.60987450182671263655638784436, 1.83342274361543785051074145781, 2.78268952173905654153624522446, 3.27991297599157474048785793937, 3.39480704296945405252045678404, 3.97055997904480693761724235852, 4.29972863374502112967357765166, 4.50891841420106988357796324547, 4.87015517522962115371896364688, 5.02882779130563354887572436271, 5.18600007951943416347800487239, 5.70941848086089266445339497339, 5.85041551936445313113891560737, 6.42725474082146945134677209836, 6.74762410876947136090532186280, 6.83291220485538776692749363331, 7.28601238535297850720143320124, 7.34284845134376802528979596671, 7.86157264842483511298107402869, 7.926355947701997810716764479918, 8.237881899898414674359498437509, 8.388878055674106088832302406700, 8.972932572654809180346407228696, 9.013758836363201580201155147901
