| L(s) = 1 | + 12·7-s − 24·17-s + 12·23-s + 14·25-s − 24·47-s + 68·49-s + 12·71-s + 20·73-s − 36·79-s + 24·89-s + 16·97-s − 24·103-s − 24·113-s − 288·119-s + 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 144·161-s + 163-s + 167-s + 34·169-s + 173-s + ⋯ |
| L(s) = 1 | + 4.53·7-s − 5.82·17-s + 2.50·23-s + 14/5·25-s − 3.50·47-s + 68/7·49-s + 1.42·71-s + 2.34·73-s − 4.05·79-s + 2.54·89-s + 1.62·97-s − 2.36·103-s − 2.25·113-s − 26.4·119-s + 1.81·121-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 + 0.0798·157-s + 11.3·161-s + 0.0783·163-s + 0.0773·167-s + 2.61·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.238151722\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.238151722\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( ( 1 - 7 T^{2} + p^{2} T^{4} )^{2} \) | 4.5.a_ao_a_dv |
| 7 | $D_{4}$ | \( ( 1 - 6 T + 20 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.7.am_cy_amm_bmo |
| 11 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 234 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) | 4.11.a_au_a_ja |
| 13 | $C_2^2$ | \( ( 1 - 17 T^{2} + p^{2} T^{4} )^{2} \) | 4.13.a_abi_a_yd |
| 17 | $D_{4}$ | \( ( 1 + 12 T + 67 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) | 4.17.y_ks_czo_otf |
| 19 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 714 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) | 4.19.a_au_a_bbm |
| 23 | $D_{4}$ | \( ( 1 - 6 T + 52 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.23.am_fk_abiq_iak |
| 29 | $D_4\times C_2$ | \( 1 - 38 T^{2} + 1611 T^{4} - 38 p^{2} T^{6} + p^{4} T^{8} \) | 4.29.a_abm_a_cjz |
| 31 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) | 4.31.a_dw_a_goc |
| 37 | $D_4\times C_2$ | \( 1 - 106 T^{2} + 5115 T^{4} - 106 p^{2} T^{6} + p^{4} T^{8} \) | 4.37.a_aec_a_hot |
| 41 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) | 4.41.a_fk_a_mfu |
| 43 | $D_4\times C_2$ | \( 1 - 116 T^{2} + 6954 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \) | 4.43.a_aem_a_khm |
| 47 | $D_{4}$ | \( ( 1 + 12 T + 82 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) | 4.47.y_lw_epc_bkne |
| 53 | $D_4\times C_2$ | \( 1 - 44 T^{2} - 810 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) | 4.53.a_abs_a_abfe |
| 59 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 10134 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) | 4.59.a_afk_a_ozu |
| 61 | $D_4\times C_2$ | \( 1 - 202 T^{2} + 17211 T^{4} - 202 p^{2} T^{6} + p^{4} T^{8} \) | 4.61.a_ahu_a_zlz |
| 67 | $D_4\times C_2$ | \( 1 - 212 T^{2} + 20106 T^{4} - 212 p^{2} T^{6} + p^{4} T^{8} \) | 4.67.a_aie_a_bdti |
| 71 | $D_{4}$ | \( ( 1 - 6 T + 148 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.71.am_mu_adxc_ccww |
| 73 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{4} \) | 4.73.au_ra_ahfs_dcqd |
| 79 | $D_{4}$ | \( ( 1 + 18 T + 212 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \) | 4.79.bk_bcu_pmy_gerq |
| 83 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) | 4.83.a_aem_a_zji |
| 89 | $D_{4}$ | \( ( 1 - 12 T + 187 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) | 4.89.ay_ty_ajuu_ejcd |
| 97 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) | 4.97.aq_sq_ahgy_ehlm |
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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
−5.85647632336576310331453000070, −5.31871271597306021886102280585, −5.06995154128883121947548966783, −5.02120092724679078633568081370, −5.01267277764484137127771885455, −4.82926431375109203657505632617, −4.70864902531554780635776175231, −4.58313693463091628521382518240, −4.39160275731996004107986256589, −4.19520886639505228898143201949, −4.07231495013921461146069993860, −3.68287300978780384499883467874, −3.32934385239527753322842380587, −3.17587429762282676042048294076, −2.67316722674796989028401179626, −2.62332648909728417249106604780, −2.62254584059497085429248051432, −1.97411812066002956016493837606, −1.95499664056508976699163920709, −1.83441676756556250183372732591, −1.72247378872722450750951894180, −1.25780254971609048363311155652, −1.03501879130440422883097462298, −0.75338981724109321410667154149, −0.24541452351697305752618739757,
0.24541452351697305752618739757, 0.75338981724109321410667154149, 1.03501879130440422883097462298, 1.25780254971609048363311155652, 1.72247378872722450750951894180, 1.83441676756556250183372732591, 1.95499664056508976699163920709, 1.97411812066002956016493837606, 2.62254584059497085429248051432, 2.62332648909728417249106604780, 2.67316722674796989028401179626, 3.17587429762282676042048294076, 3.32934385239527753322842380587, 3.68287300978780384499883467874, 4.07231495013921461146069993860, 4.19520886639505228898143201949, 4.39160275731996004107986256589, 4.58313693463091628521382518240, 4.70864902531554780635776175231, 4.82926431375109203657505632617, 5.01267277764484137127771885455, 5.02120092724679078633568081370, 5.06995154128883121947548966783, 5.31871271597306021886102280585, 5.85647632336576310331453000070
