| L(s) = 1 | + 6·9-s + 6·13-s − 2·17-s + 6·25-s + 10·29-s + 6·37-s + 30·41-s − 14·49-s − 28·53-s − 10·61-s + 9·81-s − 2·101-s + 14·113-s + 36·117-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 12·153-s + 157-s + 163-s + 167-s + 13·169-s + 173-s + ⋯ |
| L(s) = 1 | + 2·9-s + 1.66·13-s − 0.485·17-s + 6/5·25-s + 1.85·29-s + 0.986·37-s + 4.68·41-s − 2·49-s − 3.84·53-s − 1.28·61-s + 81-s − 0.199·101-s + 1.31·113-s + 3.32·117-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 0.970·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.636013848\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.636013848\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 13 | $C_2^2$ | \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) | |
| good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )^{2}( 1 + p T + p T^{2} )^{2} \) | 4.3.a_ag_a_bb |
| 5 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} )( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} ) \) | 4.5.a_ag_a_l |
| 7 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) | 4.7.a_o_a_fr |
| 11 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) | 4.11.a_w_a_nz |
| 17 | $C_2$$\times$$C_2^2$ | \( ( 1 + 2 T + p T^{2} )^{2}( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} ) \) | 4.17.c_r_adq_ahg |
| 19 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) | 4.19.a_bm_a_bpr |
| 23 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) | 4.23.a_abu_a_cjb |
| 29 | $C_2$$\times$$C_2^2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + 71 T^{2} + 10 p T^{3} + p^{2} T^{4} ) \) | 4.29.ak_bd_afa_bya |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.31.a_aeu_a_inu |
| 37 | $C_2$$\times$$C_2^2$ | \( ( 1 - 2 T + p T^{2} )^{2}( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} ) \) | 4.37.ag_cb_ajm_bdc |
| 41 | $C_2$$\times$$C_2^2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 - 10 T + 59 T^{2} - 10 p T^{3} + p^{2} T^{4} ) \) | 4.41.abe_qz_aggs_btdc |
| 43 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) | 4.43.a_adi_a_ifj |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.47.a_ahg_a_tpu |
| 53 | $C_2^2$ | \( ( 1 + 14 T + 143 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) | 4.53.bc_so_idc_crhr |
| 59 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) | 4.59.a_eo_a_plr |
| 61 | $C_2$$\times$$C_2^2$ | \( ( 1 + 10 T + p T^{2} )^{2}( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} ) \) | 4.61.k_cj_abfy_amhg |
| 67 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) | 4.67.a_fe_a_txz |
| 71 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) | 4.71.a_fm_a_wjr |
| 73 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 6 T - 37 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 6 T - 37 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \) | 4.73.a_aeg_a_kal |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.79.a_me_a_cdkg |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.83.a_amu_a_cjdu |
| 89 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 10 T + 11 T^{2} - 10 p T^{3} + p^{2} T^{4} )( 1 + 10 T + 11 T^{2} + 10 p T^{3} + p^{2} T^{4} ) \) | 4.89.a_ada_a_acsr |
| 97 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 18 T + 227 T^{2} - 18 p T^{3} + p^{2} T^{4} )( 1 + 18 T + 227 T^{2} + 18 p T^{3} + p^{2} T^{4} ) \) | 4.97.a_fa_a_lcd |
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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
−8.169929807615416160718261935040, −7.74059588786200704492120809177, −7.59855117163470967035620267316, −7.48588712046640311094704436085, −7.29219810235253796228342838475, −6.62480667182633965886027969482, −6.53805630303022501468455193145, −6.53462858180101094425206049594, −6.17720038588316100483562527121, −6.07599093742603954502546340930, −5.56195258985323777672531005211, −5.36384947418099096545841531307, −4.87759031107550079982563514708, −4.51408967744472956544458037703, −4.39180383560068879847292043416, −4.29954025928313007303675151779, −4.17479591420528510662119255991, −3.40044887374521151770588375544, −3.17726959084472836402332748000, −3.04513201793027065213653358064, −2.54841385877402980666484322557, −2.03250315268136013875504077188, −1.56427351035748475399774000430, −1.17268683819402076011014987973, −0.873890291779436135434963020463,
0.873890291779436135434963020463, 1.17268683819402076011014987973, 1.56427351035748475399774000430, 2.03250315268136013875504077188, 2.54841385877402980666484322557, 3.04513201793027065213653358064, 3.17726959084472836402332748000, 3.40044887374521151770588375544, 4.17479591420528510662119255991, 4.29954025928313007303675151779, 4.39180383560068879847292043416, 4.51408967744472956544458037703, 4.87759031107550079982563514708, 5.36384947418099096545841531307, 5.56195258985323777672531005211, 6.07599093742603954502546340930, 6.17720038588316100483562527121, 6.53462858180101094425206049594, 6.53805630303022501468455193145, 6.62480667182633965886027969482, 7.29219810235253796228342838475, 7.48588712046640311094704436085, 7.59855117163470967035620267316, 7.74059588786200704492120809177, 8.169929807615416160718261935040
