| L(s) = 1 | − 2·2-s − 2·3-s + 3·4-s − 2·5-s + 4·6-s − 2·8-s + 9-s + 4·10-s − 6·12-s − 8·13-s + 4·15-s + 4·17-s − 2·18-s − 4·19-s − 6·20-s + 4·23-s + 4·24-s + 25-s + 16·26-s + 2·27-s + 24·29-s − 8·30-s + 4·31-s + 6·32-s − 8·34-s + 3·36-s − 4·37-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 3/2·4-s − 0.894·5-s + 1.63·6-s − 0.707·8-s + 1/3·9-s + 1.26·10-s − 1.73·12-s − 2.21·13-s + 1.03·15-s + 0.970·17-s − 0.471·18-s − 0.917·19-s − 1.34·20-s + 0.834·23-s + 0.816·24-s + 1/5·25-s + 3.13·26-s + 0.384·27-s + 4.45·29-s − 1.46·30-s + 0.718·31-s + 1.06·32-s − 1.37·34-s + 1/2·36-s − 0.657·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9876999471\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9876999471\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) | |
| 5 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) | |
| 7 | | \( 1 \) | |
| good | 2 | $D_4\times C_2$ | \( 1 + p T + T^{2} - p T^{3} - 3 T^{4} - p^{2} T^{5} + p^{2} T^{6} + p^{4} T^{7} + p^{4} T^{8} \) | 4.2.c_b_ac_ad |
| 11 | $C_2^3$ | \( 1 - 14 T^{2} + 75 T^{4} - 14 p^{2} T^{6} + p^{4} T^{8} \) | 4.11.a_ao_a_cx |
| 13 | $D_{4}$ | \( ( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.13.i_ci_ku_bvq |
| 17 | $C_4\times C_2$ | \( 1 - 4 T + 10 T^{2} + 112 T^{3} - 525 T^{4} + 112 p T^{5} + 10 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.17.ae_k_ei_auf |
| 19 | $D_4\times C_2$ | \( 1 + 4 T - 18 T^{2} - 16 T^{3} + 491 T^{4} - 16 p T^{5} - 18 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.19.e_as_aq_sx |
| 23 | $D_4\times C_2$ | \( 1 - 4 T - 2 T^{2} + 112 T^{3} - 573 T^{4} + 112 p T^{5} - 2 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.23.ae_ac_ei_awb |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) | 4.29.ay_mu_aejo_bbxu |
| 31 | $C_4\times C_2$ | \( 1 - 4 T + 22 T^{2} + 272 T^{3} - 1421 T^{4} + 272 p T^{5} + 22 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.31.ae_w_km_accr |
| 37 | $D_4\times C_2$ | \( 1 + 4 T - 30 T^{2} - 112 T^{3} + 155 T^{4} - 112 p T^{5} - 30 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.37.e_abe_aei_fz |
| 41 | $C_4$ | \( ( 1 - 12 T + 86 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) | 4.41.ay_me_aeng_bhjy |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) | 4.43.abg_vk_ajdo_ctik |
| 47 | $C_2^3$ | \( 1 - 62 T^{2} + 1635 T^{4} - 62 p^{2} T^{6} + p^{4} T^{8} \) | 4.47.a_ack_a_ckx |
| 53 | $C_2^3$ | \( 1 - 34 T^{2} - 1653 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \) | 4.53.a_abi_a_aclp |
| 59 | $D_4\times C_2$ | \( 1 + 16 T + 106 T^{2} + 512 T^{3} + 3915 T^{4} + 512 p T^{5} + 106 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) | 4.59.q_ec_ts_fup |
| 61 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) | 4.61.a_aes_a_qnj |
| 67 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) | 4.67.a_afe_a_txz |
| 71 | $C_2^2$ | \( ( 1 + 134 T^{2} + p^{2} T^{4} )^{2} \) | 4.71.a_ki_a_bpmk |
| 73 | $D_4\times C_2$ | \( 1 - 28 T + 450 T^{2} - 5264 T^{3} + 48995 T^{4} - 5264 p T^{5} + 450 p^{2} T^{6} - 28 p^{3} T^{7} + p^{4} T^{8} \) | 4.73.abc_ri_ahum_cuml |
| 79 | $C_2^2$ | \( ( 1 - 8 T - 15 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) | 4.79.aq_bi_abnk_bhtr |
| 83 | $D_{4}$ | \( ( 1 - 8 T + 150 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) | 4.83.aq_oa_afnk_crjy |
| 89 | $C_4\times C_2$ | \( 1 + 12 T + 58 T^{2} - 1104 T^{3} - 13341 T^{4} - 1104 p T^{5} + 58 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.89.m_cg_abqm_attd |
| 97 | $D_{4}$ | \( ( 1 + 4 T + 126 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.97.i_ki_cqq_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
−7.60793674658602293030458647321, −7.21410739840868236413808734438, −7.21212205618006407902763413819, −6.86754595638252092013681917359, −6.49443956754981893308145471806, −6.30518875349061652825562567233, −6.19133515535854095573213963610, −6.04488003576004011846373068838, −5.69145544353189354320740457565, −5.31247106158484299285649387837, −4.97136097621529448074087384787, −4.76760356649398499101421386961, −4.64949133199915553946549913601, −4.48498073078812778223524006802, −4.07750684118881724184424322777, −3.91098114941683266328031648300, −3.36131954724444709751343913716, −2.83761919609212412368209647588, −2.68234368547621746988347378805, −2.46251070980514592966008492523, −2.45220152870654412987479986272, −1.67471952575977858278657355462, −0.878063798946793224939747534544, −0.799396323682762949938810662026, −0.68101102281013593709002747963,
0.68101102281013593709002747963, 0.799396323682762949938810662026, 0.878063798946793224939747534544, 1.67471952575977858278657355462, 2.45220152870654412987479986272, 2.46251070980514592966008492523, 2.68234368547621746988347378805, 2.83761919609212412368209647588, 3.36131954724444709751343913716, 3.91098114941683266328031648300, 4.07750684118881724184424322777, 4.48498073078812778223524006802, 4.64949133199915553946549913601, 4.76760356649398499101421386961, 4.97136097621529448074087384787, 5.31247106158484299285649387837, 5.69145544353189354320740457565, 6.04488003576004011846373068838, 6.19133515535854095573213963610, 6.30518875349061652825562567233, 6.49443956754981893308145471806, 6.86754595638252092013681917359, 7.21212205618006407902763413819, 7.21410739840868236413808734438, 7.60793674658602293030458647321
