| L(s) = 1 | + 4·4-s − 2·7-s − 12·13-s + 4·16-s + 6·19-s + 4·25-s − 8·28-s + 8·37-s + 14·43-s − 11·49-s − 48·52-s − 16·64-s + 8·67-s + 42·73-s + 24·76-s − 16·79-s + 24·91-s − 30·97-s + 16·100-s − 48·103-s − 34·109-s − 8·112-s + 10·121-s + ⋯ |
| L(s) = 1 | + 2·4-s − 0.755·7-s − 3.32·13-s + 16-s + 1.37·19-s + 4/5·25-s − 1.51·28-s + 1.31·37-s + 2.13·43-s − 1.57·49-s − 6.65·52-s − 2·64-s + 0.977·67-s + 4.91·73-s + 2.75·76-s − 1.80·79-s + 2.51·91-s − 3.04·97-s + 8/5·100-s − 4.72·103-s − 3.25·109-s − 0.755·112-s + 0.909·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.962349578\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.962349578\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) | |
| good | 2 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) | 4.2.a_ae_a_m |
| 5 | $C_2^3$ | \( 1 - 4 T^{2} - 9 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) | 4.5.a_ae_a_aj |
| 11 | $C_2^3$ | \( 1 - 10 T^{2} - 21 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} \) | 4.11.a_ak_a_av |
| 13 | $C_2^2$ | \( ( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.13.m_di_ro_cvb |
| 17 | $C_2^3$ | \( 1 - 28 T^{2} + 495 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} \) | 4.17.a_abc_a_tb |
| 19 | $C_2^2$ | \( ( 1 - 3 T + 22 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) | 4.19.ag_cb_ajm_cho |
| 23 | $C_2^3$ | \( 1 - 4 T^{2} - 513 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) | 4.23.a_ae_a_att |
| 29 | $C_2^3$ | \( 1 + 56 T^{2} + 2295 T^{4} + 56 p^{2} T^{6} + p^{4} T^{8} \) | 4.29.a_ce_a_dkh |
| 31 | $C_2^2$ | \( ( 1 - 35 T^{2} + p^{2} T^{4} )^{2} \) | 4.31.a_acs_a_erb |
| 37 | $C_2^2$ | \( ( 1 - 4 T - 21 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.37.ai_aba_aey_glv |
| 41 | $C_2^3$ | \( 1 - 76 T^{2} + 4095 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) | 4.41.a_acy_a_gbn |
| 43 | $C_2^2$ | \( ( 1 - 7 T + 6 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) | 4.43.ao_cj_abak_lts |
| 47 | $C_2^2$ | \( ( 1 + 40 T^{2} + p^{2} T^{4} )^{2} \) | 4.47.a_dc_a_ixm |
| 53 | $C_2^3$ | \( 1 + 104 T^{2} + 8007 T^{4} + 104 p^{2} T^{6} + p^{4} T^{8} \) | 4.53.a_ea_a_lvz |
| 59 | $C_2^2$ | \( ( 1 - 98 T^{2} + p^{2} T^{4} )^{2} \) | 4.59.a_aho_a_yne |
| 61 | $C_2^2$ | \( ( 1 - 95 T^{2} + p^{2} T^{4} )^{2} \) | 4.61.a_ahi_a_yjj |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) | 4.67.ai_lg_aclc_bsqg |
| 71 | $C_2^2$ | \( ( 1 - 134 T^{2} + p^{2} T^{4} )^{2} \) | 4.71.a_aki_a_bpmk |
| 73 | $C_2^2$ | \( ( 1 - 21 T + 220 T^{2} - 21 p T^{3} + p^{2} T^{4} )^{2} \) | 4.73.abq_bhx_asfi_hapw |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) | 4.79.q_pw_fzs_dafm |
| 83 | $C_2^3$ | \( 1 - 70 T^{2} - 1989 T^{4} - 70 p^{2} T^{6} + p^{4} T^{8} \) | 4.83.a_acs_a_acyn |
| 89 | $C_2^3$ | \( 1 - 172 T^{2} + 21663 T^{4} - 172 p^{2} T^{6} + p^{4} T^{8} \) | 4.89.a_agq_a_bgbf |
| 97 | $C_2^2$ | \( ( 1 + 15 T + 172 T^{2} + 15 p T^{3} + p^{2} T^{4} )^{2} \) | 4.97.be_vx_lyk_fgem |
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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
−7.65449788536869025199037440069, −7.28611164135553962048293351991, −7.23938341476161106333822337431, −7.16071019893920159551175394112, −6.86722108991159562737303464292, −6.57966205185722453803474478062, −6.38734543850391351527090635171, −6.29487843105886636490708505826, −5.88177860252042101892499556502, −5.47634578148818329906594308784, −5.23524386206974738294839835672, −5.19206327995069449432370197864, −4.88106935038421348228271750390, −4.54084991721560629772682659276, −4.14366752115068828070160686679, −3.98083562529850599305514097820, −3.54661547232924279545397885933, −3.08809857788423487352380225349, −2.68999732083173720561764359203, −2.61837785940878358292673094747, −2.55330694987324298379402633475, −2.27285973238199832523499614362, −1.58225218577514005388642311603, −1.27422423221027479082958074059, −0.39331880572850262182114037890,
0.39331880572850262182114037890, 1.27422423221027479082958074059, 1.58225218577514005388642311603, 2.27285973238199832523499614362, 2.55330694987324298379402633475, 2.61837785940878358292673094747, 2.68999732083173720561764359203, 3.08809857788423487352380225349, 3.54661547232924279545397885933, 3.98083562529850599305514097820, 4.14366752115068828070160686679, 4.54084991721560629772682659276, 4.88106935038421348228271750390, 5.19206327995069449432370197864, 5.23524386206974738294839835672, 5.47634578148818329906594308784, 5.88177860252042101892499556502, 6.29487843105886636490708505826, 6.38734543850391351527090635171, 6.57966205185722453803474478062, 6.86722108991159562737303464292, 7.16071019893920159551175394112, 7.23938341476161106333822337431, 7.28611164135553962048293351991, 7.65449788536869025199037440069
