| L(s) = 1 | + 8·11-s − 24·19-s + 2·25-s + 16·41-s + 24·49-s − 8·59-s − 18·81-s + 8·89-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 12·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s − 192·209-s + ⋯ |
| L(s) = 1 | + 2.41·11-s − 5.50·19-s + 2/5·25-s + 2.49·41-s + 24/7·49-s − 1.04·59-s − 2·81-s + 0.847·89-s − 0.363·121-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 + 0.0783·163-s + 0.0773·167-s − 0.923·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s − 13.2·209-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.066457562\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.066457562\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) | |
| good | 3 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) | 4.3.a_a_a_s |
| 7 | $C_2^2$ | \( ( 1 - 12 T^{2} + p^{2} T^{4} )^{2} \) | 4.7.a_ay_a_ji |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) | 4.11.ai_cq_alk_bww |
| 13 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) | 4.13.a_m_a_ok |
| 17 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) | 4.17.a_au_a_bac |
| 19 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) | 4.19.y_lg_dhw_rgw |
| 23 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) | 4.23.a_i_a_bpi |
| 29 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) | 4.29.a_u_a_cqo |
| 31 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) | 4.31.a_bc_a_ddm |
| 37 | $C_2^2$ | \( ( 1 - 66 T^{2} + p^{2} T^{4} )^{2} \) | 4.37.a_afc_a_kmw |
| 41 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) | 4.41.aq_ka_adho_bayo |
| 43 | $C_2^2$ | \( ( 1 - 80 T^{2} + p^{2} T^{4} )^{2} \) | 4.43.a_age_a_oyk |
| 47 | $C_2^2$ | \( ( 1 - 76 T^{2} + p^{2} T^{4} )^{2} \) | 4.47.a_afw_a_pcc |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.53.a_aie_a_yyg |
| 59 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) | 4.59.i_ka_cds_bjcw |
| 61 | $C_2^2$ | \( ( 1 + 110 T^{2} + p^{2} T^{4} )^{2} \) | 4.61.a_im_a_bcxq |
| 67 | $C_2^2$ | \( ( 1 - 128 T^{2} + p^{2} T^{4} )^{2} \) | 4.67.a_ajw_a_blnm |
| 71 | $C_2^2$ | \( ( 1 + 94 T^{2} + p^{2} T^{4} )^{2} \) | 4.71.a_hg_a_bbzq |
| 73 | $C_2^2$ | \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{2} \) | 4.73.a_ajk_a_bluk |
| 79 | $C_2^2$ | \( ( 1 + 110 T^{2} + p^{2} T^{4} )^{2} \) | 4.79.a_im_a_bkjm |
| 83 | $C_2^2$ | \( ( 1 - 16 T^{2} + p^{2} T^{4} )^{2} \) | 4.83.a_abg_a_utu |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) | 4.89.ai_oq_adfk_cyqw |
| 97 | $C_2^2$ | \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \) | 4.97.a_bs_a_bcok |
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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
−6.68732966563527244933889782197, −6.64449408518991895737168517260, −6.59307601125213246059368864542, −6.17501312662379344170529761270, −6.09794092419407075371583347285, −5.97448181530948798255161037722, −5.74689493350658031766562918669, −5.45008801933373732200317612016, −5.10984353791789850908754551248, −4.67182813564779320255837712074, −4.43932208403193096701780525197, −4.32572139969945155905295544977, −4.21837703320805522681484836290, −4.07252251090164131964052978839, −3.74567981292682963658515094282, −3.66288589633197658939459625567, −3.09335588819717226241523114941, −2.65220906196257188783399170783, −2.60892764683498982900389391637, −2.25742738010312611172239463610, −1.86393181669072006771440172476, −1.78405238580127197602514369530, −1.33548764282829712681934600340, −0.73097470812397957505878089683, −0.44789852922513516232191787121,
0.44789852922513516232191787121, 0.73097470812397957505878089683, 1.33548764282829712681934600340, 1.78405238580127197602514369530, 1.86393181669072006771440172476, 2.25742738010312611172239463610, 2.60892764683498982900389391637, 2.65220906196257188783399170783, 3.09335588819717226241523114941, 3.66288589633197658939459625567, 3.74567981292682963658515094282, 4.07252251090164131964052978839, 4.21837703320805522681484836290, 4.32572139969945155905295544977, 4.43932208403193096701780525197, 4.67182813564779320255837712074, 5.10984353791789850908754551248, 5.45008801933373732200317612016, 5.74689493350658031766562918669, 5.97448181530948798255161037722, 6.09794092419407075371583347285, 6.17501312662379344170529761270, 6.59307601125213246059368864542, 6.64449408518991895737168517260, 6.68732966563527244933889782197
