| L(s) = 1 | − 4·3-s − 2·9-s + 12·11-s + 6·25-s + 40·27-s − 48·33-s + 36·59-s − 12·67-s − 24·75-s − 55·81-s + 12·89-s − 12·97-s − 24·99-s + 12·113-s + 86·121-s + ⋯ |
| L(s) = 1 | − 2.30·3-s − 2/3·9-s + 3.61·11-s + 6/5·25-s + 7.69·27-s − 8.35·33-s + 4.68·59-s − 1.46·67-s − 2.77·75-s − 6.11·81-s + 1.27·89-s − 1.21·97-s − 2.41·99-s + 1.12·113-s + 7.81·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7940059667\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7940059667\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) | |
| good | 3 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) | 4.3.e_s_bo_dn |
| 5 | $C_2^2$ | \( ( 1 - 3 T^{2} + p^{2} T^{4} )^{2} \) | 4.5.a_ag_a_ch |
| 7 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) | 4.7.a_a_a_du |
| 13 | $C_2^2$ | \( ( 1 + 12 T^{2} + p^{2} T^{4} )^{2} \) | 4.13.a_y_a_so |
| 17 | $C_2^2$ | \( ( 1 - 16 T^{2} + p^{2} T^{4} )^{2} \) | 4.17.a_abg_a_bgc |
| 19 | $C_2^2$ | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \) | 4.19.a_abo_a_bre |
| 23 | $C_2^2$ | \( ( 1 - 39 T^{2} + p^{2} T^{4} )^{2} \) | 4.23.a_ada_a_dvf |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.29.a_em_a_hmc |
| 31 | $C_2^2$ | \( ( 1 + T^{2} + p^{2} T^{4} )^{2} \) | 4.31.a_c_a_cvz |
| 37 | $C_2^2$ | \( ( 1 - 11 T^{2} + p^{2} T^{4} )^{2} \) | 4.37.a_aw_a_efz |
| 41 | $C_2^2$ | \( ( 1 - 80 T^{2} + p^{2} T^{4} )^{2} \) | 4.41.a_age_a_olm |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \) | 4.43.a_abc_a_ftu |
| 47 | $C_2^2$ | \( ( 1 + 18 T^{2} + p^{2} T^{4} )^{2} \) | 4.47.a_bk_a_hak |
| 53 | $C_2^2$ | \( ( 1 - 78 T^{2} + p^{2} T^{4} )^{2} \) | 4.53.a_aga_a_ric |
| 59 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{4} \) | 4.59.abk_bbu_antg_evlj |
| 61 | $C_2^2$ | \( ( 1 - 102 T^{2} + p^{2} T^{4} )^{2} \) | 4.61.a_ahw_a_bakk |
| 67 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) | 4.67.m_mk_dsy_byrj |
| 71 | $C_2^2$ | \( ( 1 - 135 T^{2} + p^{2} T^{4} )^{2} \) | 4.71.a_akk_a_bpwt |
| 73 | $C_2^2$ | \( ( 1 + 16 T^{2} + p^{2} T^{4} )^{2} \) | 4.73.a_bg_a_qdu |
| 79 | $C_2^2$ | \( ( 1 + 102 T^{2} + p^{2} T^{4} )^{2} \) | 4.79.a_hw_a_bhwg |
| 83 | $C_2^2$ | \( ( 1 - 94 T^{2} + p^{2} T^{4} )^{2} \) | 4.83.a_ahg_a_bhlu |
| 89 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{4} \) | 4.89.am_pu_aexk_dgqt |
| 97 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) | 4.97.m_ra_fim_dvdj |
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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
−6.79536074187770751502483974123, −6.52474972609800433504615115715, −6.26422680904900706901010096774, −6.20078051303962782123265696224, −5.99470287866265457385855065188, −5.76397937741583149512733294224, −5.58349190749881099487207800591, −5.40656983142375581585199279131, −5.11235572518218366712228285518, −4.87713797124431501872150048285, −4.65955993797678368310193221436, −4.60821082215372322020211169163, −4.11189990760571803221091746426, −3.73338730687452855085517830861, −3.71463127766512015223141410463, −3.48303834320109886930984411656, −3.21943918016547184858626268240, −2.62365286681156762615871316533, −2.54269439678761374884913393986, −2.43527840288235998710928354581, −1.73842368189064668397107007969, −1.30017353600547342961493304216, −0.971387106987741602078171056593, −0.867011025221689155993559846267, −0.28929086124039802173321070482,
0.28929086124039802173321070482, 0.867011025221689155993559846267, 0.971387106987741602078171056593, 1.30017353600547342961493304216, 1.73842368189064668397107007969, 2.43527840288235998710928354581, 2.54269439678761374884913393986, 2.62365286681156762615871316533, 3.21943918016547184858626268240, 3.48303834320109886930984411656, 3.71463127766512015223141410463, 3.73338730687452855085517830861, 4.11189990760571803221091746426, 4.60821082215372322020211169163, 4.65955993797678368310193221436, 4.87713797124431501872150048285, 5.11235572518218366712228285518, 5.40656983142375581585199279131, 5.58349190749881099487207800591, 5.76397937741583149512733294224, 5.99470287866265457385855065188, 6.20078051303962782123265696224, 6.26422680904900706901010096774, 6.52474972609800433504615115715, 6.79536074187770751502483974123
