| L(s) = 1 | + 12·11-s + 20·25-s − 16·49-s − 24·59-s − 32·67-s − 24·89-s − 32·97-s + 24·113-s + 86·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | + 3.61·11-s + 4·25-s − 2.28·49-s − 3.12·59-s − 3.90·67-s − 2.54·89-s − 3.24·97-s + 2.25·113-s + 7.81·121-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 + 0.0783·163-s + 0.0773·167-s − 0.307·169-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 + 0.0688·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.741173703\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.741173703\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) | |
| good | 5 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.5.a_au_a_fu |
| 7 | $C_2^2$ | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{2} \) | 4.7.a_q_a_gg |
| 13 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) | 4.13.a_e_a_ne |
| 17 | $C_2^2$ | \( ( 1 + 16 T^{2} + p^{2} T^{4} )^{2} \) | 4.17.a_bg_a_bgc |
| 19 | $C_2^2$ | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \) | 4.19.a_abo_a_bre |
| 23 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) | 4.23.a_e_a_bow |
| 29 | $C_2^2$ | \( ( 1 + 52 T^{2} + p^{2} T^{4} )^{2} \) | 4.29.a_ea_a_gms |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) | 4.31.a_do_a_fzi |
| 37 | $C_2^2$ | \( ( 1 - 62 T^{2} + p^{2} T^{4} )^{2} \) | 4.37.a_aeu_a_jte |
| 41 | $C_2^2$ | \( ( 1 - 32 T^{2} + p^{2} T^{4} )^{2} \) | 4.41.a_acm_a_gms |
| 43 | $C_2^2$ | \( ( 1 - 68 T^{2} + p^{2} T^{4} )^{2} \) | 4.43.a_afg_a_mic |
| 47 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) | 4.47.a_bc_a_gvm |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.53.a_aie_a_yyg |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) | 4.59.y_rk_hoq_csnm |
| 61 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) | 4.61.a_ca_a_mag |
| 67 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) | 4.67.bg_zc_moe_esao |
| 71 | $C_2^2$ | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \) | 4.71.a_aka_a_bnxu |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.73.a_alg_a_bvhu |
| 79 | $C_2^2$ | \( ( 1 + 152 T^{2} + p^{2} T^{4} )^{2} \) | 4.79.a_ls_a_caqs |
| 83 | $C_2^2$ | \( ( 1 - 134 T^{2} + p^{2} T^{4} )^{2} \) | 4.83.a_aki_a_buyo |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) | 4.89.y_wa_kts_ezco |
| 97 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) | 4.97.bg_bds_quy_hruc |
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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
−6.20830582254612343383157933785, −6.08130260710433938326395289865, −5.71848071561613727451902823187, −5.63226615351068712772797279640, −5.47449020959184570641406874895, −4.86404271279439900342699138001, −4.81876499274502794905529308995, −4.70571902775812438424484987148, −4.50766302391888223956339675707, −4.42806329099519539513142770787, −4.01365294742535125480792458361, −4.00725900276259756110602620704, −3.64372696628676620261023708599, −3.17758220875247705377730357964, −3.15848068301747871904406934433, −3.14001047906165637850980433751, −2.81462260756927745507567055073, −2.67192848873848027663646687910, −1.86291904828081536791279261673, −1.83228422071514076277895722157, −1.68039122167709311181748731305, −1.21226423789033671233287779348, −1.06515969058692050244643648720, −1.02357803281580801932460598546, −0.21640058606759765525414326375,
0.21640058606759765525414326375, 1.02357803281580801932460598546, 1.06515969058692050244643648720, 1.21226423789033671233287779348, 1.68039122167709311181748731305, 1.83228422071514076277895722157, 1.86291904828081536791279261673, 2.67192848873848027663646687910, 2.81462260756927745507567055073, 3.14001047906165637850980433751, 3.15848068301747871904406934433, 3.17758220875247705377730357964, 3.64372696628676620261023708599, 4.00725900276259756110602620704, 4.01365294742535125480792458361, 4.42806329099519539513142770787, 4.50766302391888223956339675707, 4.70571902775812438424484987148, 4.81876499274502794905529308995, 4.86404271279439900342699138001, 5.47449020959184570641406874895, 5.63226615351068712772797279640, 5.71848071561613727451902823187, 6.08130260710433938326395289865, 6.20830582254612343383157933785
