| L(s) = 1 | + 4-s + 2·7-s + 6·13-s − 2·25-s + 2·28-s − 18·31-s − 8·37-s + 22·43-s − 11·49-s + 6·52-s − 48·61-s − 64-s + 14·67-s − 16·79-s + 12·91-s − 24·97-s − 2·100-s + 30·103-s + 16·109-s + 14·121-s − 18·124-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 0.755·7-s + 1.66·13-s − 2/5·25-s + 0.377·28-s − 3.23·31-s − 1.31·37-s + 3.35·43-s − 1.57·49-s + 0.832·52-s − 6.14·61-s − 1/8·64-s + 1.71·67-s − 1.80·79-s + 1.25·91-s − 2.43·97-s − 1/5·100-s + 2.95·103-s + 1.53·109-s + 1.27·121-s − 1.61·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.067271434\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.067271434\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) | |
| 3 | | \( 1 \) | |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) | |
| good | 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 | $C_2^3$ | \( 1 - 14 T^{2} + 75 T^{4} - 14 p^{2} T^{6} + p^{4} T^{8} \) | 4.11.a_ao_a_cx |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2}( 1 + 2 T + p T^{2} )^{2} \) | 4.13.ag_bp_ags_bfw |
| 17 | $C_2^2$ | \( ( 1 + 31 T^{2} + p^{2} T^{4} )^{2} \) | 4.17.a_ck_a_chf |
| 19 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) | 4.19.a_u_a_bfq |
| 23 | $C_2^3$ | \( 1 + 37 T^{2} + 840 T^{4} + 37 p^{2} T^{6} + p^{4} T^{8} \) | 4.23.a_bl_a_bgi |
| 29 | $C_2^3$ | \( 1 + 49 T^{2} + 1560 T^{4} + 49 p^{2} T^{6} + p^{4} T^{8} \) | 4.29.a_bx_a_cia |
| 31 | $C_2^2$ | \( ( 1 + 9 T + 58 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) | 4.31.s_hp_cjq_pgm |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) | 4.37.i_gq_bjk_ouw |
| 41 | $C_2^3$ | \( 1 - 34 T^{2} - 525 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \) | 4.41.a_abi_a_auf |
| 43 | $C_2^2$ | \( ( 1 - 11 T + 78 T^{2} - 11 p T^{3} + p^{2} T^{4} )^{2} \) | 4.43.aw_kr_adyk_bdwm |
| 47 | $C_2^3$ | \( 1 - 46 T^{2} - 93 T^{4} - 46 p^{2} T^{6} + p^{4} T^{8} \) | 4.47.a_abu_a_adp |
| 53 | $C_2^2$ | \( ( 1 - 97 T^{2} + p^{2} T^{4} )^{2} \) | 4.53.a_ahm_a_wfz |
| 59 | $C_2^3$ | \( 1 - 43 T^{2} - 1632 T^{4} - 43 p^{2} T^{6} + p^{4} T^{8} \) | 4.59.a_abr_a_acku |
| 61 | $C_2^2$ | \( ( 1 + 24 T + 253 T^{2} + 24 p T^{3} + p^{2} T^{4} )^{2} \) | 4.61.bw_bpq_whs_ibqx |
| 67 | $C_2^2$ | \( ( 1 - 7 T - 18 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) | 4.67.ao_n_abak_xmi |
| 71 | $C_2^2$ | \( ( 1 - 133 T^{2} + p^{2} T^{4} )^{2} \) | 4.71.a_akg_a_bpcd |
| 73 | $C_2^2$ | \( ( 1 - 98 T^{2} + p^{2} T^{4} )^{2} \) | 4.73.a_aho_a_bdzi |
| 79 | $C_2^2$ | \( ( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) | 4.79.q_bi_bnk_bhtr |
| 83 | $C_2^3$ | \( 1 - 154 T^{2} + 16827 T^{4} - 154 p^{2} T^{6} + p^{4} T^{8} \) | 4.83.a_afy_a_yxf |
| 89 | $C_2^2$ | \( ( 1 + 151 T^{2} + p^{2} T^{4} )^{2} \) | 4.89.a_lq_a_cfeh |
| 97 | $C_2^2$ | \( ( 1 + 12 T + 145 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) | 4.97.y_qs_ipk_dwgx |
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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
−7.15657254596665847782693195790, −6.80065997531539907749705774911, −6.71512768256105797400079280194, −6.16339690870860772348726253597, −6.03348416015784148840690234468, −5.89348056012499520791470840751, −5.72949021301107508416401874465, −5.68162254063764288779969240681, −5.37312083424044954059245534305, −4.77835816922207746460533198812, −4.64609867260818235649262670863, −4.52054694888318123890264516016, −4.48761976047721948612286083240, −3.89192534354639520249616578488, −3.63228123526743971162753417572, −3.40662048292241576833781856455, −3.36659029838027701719742355204, −3.05184328773368890737573718275, −2.54229699463433584906642505901, −2.26197990181268747025589377655, −1.90790469890449222718143119951, −1.56603016969739666358938282100, −1.54563260237194862419751805862, −1.03389679832120234932848326167, −0.29540853119602074487161489851,
0.29540853119602074487161489851, 1.03389679832120234932848326167, 1.54563260237194862419751805862, 1.56603016969739666358938282100, 1.90790469890449222718143119951, 2.26197990181268747025589377655, 2.54229699463433584906642505901, 3.05184328773368890737573718275, 3.36659029838027701719742355204, 3.40662048292241576833781856455, 3.63228123526743971162753417572, 3.89192534354639520249616578488, 4.48761976047721948612286083240, 4.52054694888318123890264516016, 4.64609867260818235649262670863, 4.77835816922207746460533198812, 5.37312083424044954059245534305, 5.68162254063764288779969240681, 5.72949021301107508416401874465, 5.89348056012499520791470840751, 6.03348416015784148840690234468, 6.16339690870860772348726253597, 6.71512768256105797400079280194, 6.80065997531539907749705774911, 7.15657254596665847782693195790
