| L(s) = 1 | + 4·7-s + 6·9-s − 12·17-s − 2·25-s + 4·31-s − 24·41-s + 24·47-s − 18·49-s + 24·63-s − 12·71-s + 32·73-s + 17·81-s + 12·89-s + 8·97-s + 56·103-s + 24·113-s − 48·119-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 72·153-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | + 1.51·7-s + 2·9-s − 2.91·17-s − 2/5·25-s + 0.718·31-s − 3.74·41-s + 3.50·47-s − 2.57·49-s + 3.02·63-s − 1.42·71-s + 3.74·73-s + 17/9·81-s + 1.27·89-s + 0.812·97-s + 5.51·103-s + 2.25·113-s − 4.40·119-s − 0.181·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 5.82·153-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.872174599\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.872174599\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) | |
| 11 | $C_2$ | \( ( 1 + T^{2} )^{2} \) | |
| good | 3 | $D_4\times C_2$ | \( 1 - 2 p T^{2} + 19 T^{4} - 2 p^{3} T^{6} + p^{4} T^{8} \) | 4.3.a_ag_a_t |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) | 4.7.ae_bi_adk_op |
| 13 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) | 4.13.a_abk_a_zm |
| 17 | $D_{4}$ | \( ( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.17.m_di_tk_dpj |
| 19 | $C_2^2$ | \( ( 1 - 29 T^{2} + p^{2} T^{4} )^{2} \) | 4.19.a_acg_a_cid |
| 23 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) | 4.23.a_cy_a_dsg |
| 29 | $D_4\times C_2$ | \( 1 - 62 T^{2} + 1995 T^{4} - 62 p^{2} T^{6} + p^{4} T^{8} \) | 4.29.a_ack_a_cyt |
| 31 | $D_{4}$ | \( ( 1 - 2 T + 45 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.31.ae_dq_als_gfj |
| 37 | $D_4\times C_2$ | \( 1 + 30 T^{2} + 371 T^{4} + 30 p^{2} T^{6} + p^{4} T^{8} \) | 4.37.a_be_a_oh |
| 41 | $D_{4}$ | \( ( 1 + 12 T + 110 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) | 4.41.y_oa_fjk_boiw |
| 43 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) | 4.43.a_adw_a_jek |
| 47 | $D_{4}$ | \( ( 1 - 12 T + 122 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) | 4.47.ay_oy_agaa_bwpa |
| 53 | $C_2^2$ | \( ( 1 - 81 T^{2} + p^{2} T^{4} )^{2} \) | 4.53.a_agg_a_sal |
| 59 | $D_4\times C_2$ | \( 1 - 60 T^{2} + 3254 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} \) | 4.59.a_aci_a_eve |
| 61 | $D_4\times C_2$ | \( 1 - 126 T^{2} + 9611 T^{4} - 126 p^{2} T^{6} + p^{4} T^{8} \) | 4.61.a_aew_a_ofr |
| 67 | $D_4\times C_2$ | \( 1 - 132 T^{2} + 8726 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} \) | 4.67.a_afc_a_mxq |
| 71 | $D_{4}$ | \( ( 1 + 6 T + 133 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.71.m_lq_dqe_bwqt |
| 73 | $D_{4}$ | \( ( 1 - 16 T + 138 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) | 4.73.abg_um_ajzs_dvfy |
| 79 | $C_2^2$ | \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \) | 4.79.a_gq_a_bdko |
| 83 | $D_4\times C_2$ | \( 1 - 156 T^{2} + 15254 T^{4} - 156 p^{2} T^{6} + p^{4} T^{8} \) | 4.83.a_aga_a_wos |
| 89 | $D_{4}$ | \( ( 1 - 6 T + 155 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.89.am_ni_aeiq_cqlv |
| 97 | $D_{4}$ | \( ( 1 - 4 T + 126 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.97.ai_ki_acqq_cdxu |
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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
−6.64474882316490609151195575027, −6.35843432940097195410101207314, −6.26739815540197761030463565480, −6.03632164910375400343508474429, −6.02838552353601626929765787572, −5.31871478493477125571422457457, −5.20222503783342595796579401903, −4.98172965892397657334732445217, −4.80483011091446624919836874900, −4.74776518883187962357222454155, −4.51346625442701677518988686764, −4.31083377287863175179630633447, −4.03548874938181673635169338302, −3.80926093603423359570953115707, −3.43587980850040703939751006016, −3.38510541161935362393606909705, −3.09028009196249167308122332026, −2.35687436173230488484404419294, −2.35582711192882917380268986287, −2.01549919633717703976572247227, −1.89217525933675426916108779653, −1.70358214320150403022290195241, −1.15647766682159664470726411949, −0.942697990203173311108348798249, −0.29541098311385958582300745777,
0.29541098311385958582300745777, 0.942697990203173311108348798249, 1.15647766682159664470726411949, 1.70358214320150403022290195241, 1.89217525933675426916108779653, 2.01549919633717703976572247227, 2.35582711192882917380268986287, 2.35687436173230488484404419294, 3.09028009196249167308122332026, 3.38510541161935362393606909705, 3.43587980850040703939751006016, 3.80926093603423359570953115707, 4.03548874938181673635169338302, 4.31083377287863175179630633447, 4.51346625442701677518988686764, 4.74776518883187962357222454155, 4.80483011091446624919836874900, 4.98172965892397657334732445217, 5.20222503783342595796579401903, 5.31871478493477125571422457457, 6.02838552353601626929765787572, 6.03632164910375400343508474429, 6.26739815540197761030463565480, 6.35843432940097195410101207314, 6.64474882316490609151195575027
