| L(s) = 1 | + 16·23-s + 16·25-s − 48·47-s − 8·49-s + 32·71-s + 16·73-s + 8·97-s + 4·121-s + ⋯ |
| L(s) = 1 | + 3.33·23-s + 16/5·25-s − 7.00·47-s − 8/7·49-s + 3.79·71-s + 1.87·73-s + 0.812·97-s + 4/11·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.077672532\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.077672532\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 3 | | \( 1 \) | |
| good | 5 | $C_2^2$ | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \) | 4.5.a_aq_a_ek |
| 7 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) | 4.7.a_i_a_ek |
| 11 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) | 4.11.a_ae_a_jm |
| 13 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) | 4.13.a_am_a_ok |
| 17 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) | 4.17.a_am_a_xq |
| 19 | $C_2^2$ | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \) | 4.19.a_aci_a_ckk |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) | 4.23.aq_hg_acai_lpu |
| 29 | $C_2^2$ | \( ( 1 - 40 T^{2} + p^{2} T^{4} )^{2} \) | 4.29.a_adc_a_ewg |
| 31 | $C_2^2$ | \( ( 1 + 52 T^{2} + p^{2} T^{4} )^{2} \) | 4.31.a_ea_a_gvy |
| 37 | $C_2^2$ | \( ( 1 - 54 T^{2} + p^{2} T^{4} )^{2} \) | 4.37.a_aee_a_ijm |
| 41 | $C_2^2$ | \( ( 1 + 42 T^{2} + p^{2} T^{4} )^{2} \) | 4.41.a_dg_a_hpe |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \) | 4.43.a_abc_a_ftu |
| 47 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{4} \) | 4.47.bw_bom_uge_gola |
| 53 | $C_2^2$ | \( ( 1 - 56 T^{2} + p^{2} T^{4} )^{2} \) | 4.53.a_aei_a_mys |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.59.a_ajc_a_bexi |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) | 4.61.a_em_a_pzq |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.67.a_aki_a_bnvy |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) | 4.71.abg_zs_ancy_fbmc |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) | 4.73.aq_oy_afoq_cqks |
| 79 | $C_2^2$ | \( ( 1 + 148 T^{2} + p^{2} T^{4} )^{2} \) | 4.79.a_lk_a_bywo |
| 83 | $C_2^2$ | \( ( 1 - 146 T^{2} + p^{2} T^{4} )^{2} \) | 4.83.a_alg_a_bzxu |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.89.a_ns_a_cshy |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) | 4.97.ai_pw_admu_dmla |
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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
−5.75219959449486048294857606000, −5.73208639819931564203060250922, −5.32201289736398754478144323215, −5.15844971534477600173296469067, −4.90726776736756231204293871555, −4.88399755543455127204067869784, −4.82679308790130689819020674812, −4.66339528728776198337163153606, −4.61977830716439079667819380804, −4.02620889521305129903237906530, −3.65307683645201894592301801425, −3.61276147282695002581874862634, −3.39299738596720545294492703169, −3.20975549532666909201892769790, −3.13210176167670943154091329965, −2.84879326075355759078496619225, −2.54006404426775800203284672436, −2.47680254987283298954188427455, −1.98730608817679660566189929145, −1.77361137302120758915139392589, −1.40111311053621843178605339315, −1.34208639177842759899545389347, −0.878242228840374217547190646074, −0.75244088271128794053674813595, −0.28056352961963526344828105281,
0.28056352961963526344828105281, 0.75244088271128794053674813595, 0.878242228840374217547190646074, 1.34208639177842759899545389347, 1.40111311053621843178605339315, 1.77361137302120758915139392589, 1.98730608817679660566189929145, 2.47680254987283298954188427455, 2.54006404426775800203284672436, 2.84879326075355759078496619225, 3.13210176167670943154091329965, 3.20975549532666909201892769790, 3.39299738596720545294492703169, 3.61276147282695002581874862634, 3.65307683645201894592301801425, 4.02620889521305129903237906530, 4.61977830716439079667819380804, 4.66339528728776198337163153606, 4.82679308790130689819020674812, 4.88399755543455127204067869784, 4.90726776736756231204293871555, 5.15844971534477600173296469067, 5.32201289736398754478144323215, 5.73208639819931564203060250922, 5.75219959449486048294857606000
