| L(s) = 1 | + 4·4-s − 8·9-s + 4·16-s + 12·23-s − 2·25-s + 12·29-s − 32·36-s + 20·43-s − 2·49-s − 12·53-s + 24·61-s − 16·64-s + 28·79-s + 30·81-s + 48·92-s − 8·100-s − 24·101-s + 32·103-s − 24·107-s − 36·113-s + 48·116-s + 8·121-s + 127-s + 131-s + 137-s + 139-s − 32·144-s + ⋯ |
| L(s) = 1 | + 2·4-s − 8/3·9-s + 16-s + 2.50·23-s − 2/5·25-s + 2.22·29-s − 5.33·36-s + 3.04·43-s − 2/7·49-s − 1.64·53-s + 3.07·61-s − 2·64-s + 3.15·79-s + 10/3·81-s + 5.00·92-s − 4/5·100-s − 2.38·101-s + 3.15·103-s − 2.32·107-s − 3.38·113-s + 4.45·116-s + 8/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 8/3·144-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.775077125\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.775077125\) |
| \(L(\frac{3}{2})\) |
|
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 | 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) | |
| 13 | | \( 1 \) | |
| good | 2 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) | 4.2.a_ae_a_m |
| 3 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) | 4.3.a_i_a_bi |
| 5 | $C_2^3$ | \( 1 + 2 T^{2} - 21 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \) | 4.5.a_c_a_av |
| 11 | $C_2^2$ | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \) | 4.11.a_ai_a_jy |
| 17 | $C_2^2$ | \( ( 1 + 32 T^{2} + p^{2} T^{4} )^{2} \) | 4.17.a_cm_a_cjq |
| 19 | $D_4\times C_2$ | \( 1 - 22 T^{2} + 195 T^{4} - 22 p^{2} T^{6} + p^{4} T^{8} \) | 4.19.a_aw_a_hn |
| 23 | $D_{4}$ | \( ( 1 - 6 T + 47 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.23.am_fa_abgi_hhj |
| 29 | $D_{4}$ | \( ( 1 - 6 T + 59 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.29.am_fy_aboq_ksx |
| 31 | $D_4\times C_2$ | \( 1 - 86 T^{2} + 3699 T^{4} - 86 p^{2} T^{6} + p^{4} T^{8} \) | 4.31.a_adi_a_fmh |
| 37 | $D_4\times C_2$ | \( 1 - 104 T^{2} + 5154 T^{4} - 104 p^{2} T^{6} + p^{4} T^{8} \) | 4.37.a_aea_a_hqg |
| 41 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 3654 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) | 4.41.a_acy_a_fko |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{4} \) | 4.43.au_mk_aeom_bkkx |
| 47 | $D_4\times C_2$ | \( 1 - 166 T^{2} + 11235 T^{4} - 166 p^{2} T^{6} + p^{4} T^{8} \) | 4.47.a_agk_a_qqd |
| 53 | $D_{4}$ | \( ( 1 + 6 T + 107 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.53.m_jq_cvw_bexf |
| 59 | $D_4\times C_2$ | \( 1 - 100 T^{2} + 4854 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \) | 4.59.a_adw_a_hes |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) | 4.61.ay_rs_ahue_cvyc |
| 67 | $D_4\times C_2$ | \( 1 - 52 T^{2} - 714 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) | 4.67.a_aca_a_abbm |
| 71 | $D_4\times C_2$ | \( 1 - 112 T^{2} + 6018 T^{4} - 112 p^{2} T^{6} + p^{4} T^{8} \) | 4.71.a_aei_a_ixm |
| 73 | $D_4\times C_2$ | \( 1 - 206 T^{2} + 19467 T^{4} - 206 p^{2} T^{6} + p^{4} T^{8} \) | 4.73.a_ahy_a_bcut |
| 79 | $D_{4}$ | \( ( 1 - 14 T + 135 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) | 4.79.abc_ry_aiwm_dngd |
| 83 | $D_4\times C_2$ | \( 1 - 134 T^{2} + 12435 T^{4} - 134 p^{2} T^{6} + p^{4} T^{8} \) | 4.83.a_afe_a_skh |
| 89 | $D_4\times C_2$ | \( 1 - 334 T^{2} + 43659 T^{4} - 334 p^{2} T^{6} + p^{4} T^{8} \) | 4.89.a_amw_a_cmpf |
| 97 | $D_4\times C_2$ | \( 1 - 62 T^{2} + 19131 T^{4} - 62 p^{2} T^{6} + p^{4} T^{8} \) | 4.97.a_ack_a_bchv |
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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
−6.90849367139665123020828639816, −6.80593200785392358966371295315, −6.39768233294012560897383534203, −6.38809563866148815457226451948, −6.38162476015915415516721169592, −5.82586967476106075460732064030, −5.64853956109799753822711512629, −5.61711604944234924116434855354, −5.15702037904499284917634074172, −4.98004940742557123707613387922, −4.96868573668115840494075677370, −4.53016237893200899953251155818, −4.11483846106220186376339103959, −3.93484271803200795199379939235, −3.65361703743819101139162561290, −3.14887016743453032469639274128, −3.00611093591508705315434933700, −2.74061153298549914710427263892, −2.73768468888643032198507652173, −2.45600496581358883516590133258, −2.18203868392107879746775730904, −1.79555544560301329292161094314, −1.29139106747678635455856508547, −0.837244320492545906829699994381, −0.50298107073388121815470640618,
0.50298107073388121815470640618, 0.837244320492545906829699994381, 1.29139106747678635455856508547, 1.79555544560301329292161094314, 2.18203868392107879746775730904, 2.45600496581358883516590133258, 2.73768468888643032198507652173, 2.74061153298549914710427263892, 3.00611093591508705315434933700, 3.14887016743453032469639274128, 3.65361703743819101139162561290, 3.93484271803200795199379939235, 4.11483846106220186376339103959, 4.53016237893200899953251155818, 4.96868573668115840494075677370, 4.98004940742557123707613387922, 5.15702037904499284917634074172, 5.61711604944234924116434855354, 5.64853956109799753822711512629, 5.82586967476106075460732064030, 6.38162476015915415516721169592, 6.38809563866148815457226451948, 6.39768233294012560897383534203, 6.80593200785392358966371295315, 6.90849367139665123020828639816
