| L(s) = 1 | − 8·11-s − 8·29-s − 8·31-s + 32·41-s + 16·49-s + 16·59-s + 24·61-s − 8·71-s + 16·79-s + 2·81-s + 24·89-s + 40·101-s + 24·109-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 + ⋯ |
| L(s) = 1 | − 2.41·11-s − 1.48·29-s − 1.43·31-s + 4.99·41-s + 16/7·49-s + 2.08·59-s + 3.07·61-s − 0.949·71-s + 1.80·79-s + 2/9·81-s + 2.54·89-s + 3.98·101-s + 2.29·109-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.815161787\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.815161787\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| good | 3 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} ) \) | 4.3.a_a_a_ac |
| 7 | $D_4\times C_2$ | \( 1 - 16 T^{2} + 142 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) | 4.7.a_aq_a_fm |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) | 4.11.i_cq_lk_bww |
| 13 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) | 4.13.a_am_a_ok |
| 17 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) | 4.17.a_abc_a_bdu |
| 19 | $C_2^2$ | \( ( 1 + 18 T^{2} + p^{2} T^{4} )^{2} \) | 4.19.a_bk_a_bog |
| 23 | $D_4\times C_2$ | \( 1 + 16 T^{2} + 142 T^{4} + 16 p^{2} T^{6} + p^{4} T^{8} \) | 4.23.a_q_a_fm |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) | 4.29.i_fk_bca_jog |
| 31 | $C_4$ | \( ( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.31.i_ee_xs_hlm |
| 37 | $D_4\times C_2$ | \( 1 + 20 T^{2} + 1558 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} \) | 4.37.a_u_a_chy |
| 41 | $C_4$ | \( ( 1 - 16 T + 126 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) | 4.41.abg_to_ahxo_chni |
| 43 | $D_4\times C_2$ | \( 1 - 80 T^{2} + 5118 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} \) | 4.43.a_adc_a_how |
| 47 | $D_4\times C_2$ | \( 1 - 128 T^{2} + 8014 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8} \) | 4.47.a_aey_a_lwg |
| 53 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 9238 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) | 4.53.a_afk_a_nri |
| 59 | $C_4$ | \( ( 1 - 8 T + 114 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) | 4.59.aq_lg_aecm_bosc |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) | 4.61.ay_rs_ahue_cvyc |
| 67 | $D_4\times C_2$ | \( 1 - 176 T^{2} + 16542 T^{4} - 176 p^{2} T^{6} + p^{4} T^{8} \) | 4.67.a_agu_a_ymg |
| 71 | $C_4$ | \( ( 1 + 4 T + 126 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.71.i_ki_ciq_bptu |
| 73 | $C_2^2$ | \( ( 1 - 126 T^{2} + p^{2} T^{4} )^{2} \) | 4.73.a_ajs_a_bngo |
| 79 | $D_{4}$ | \( ( 1 - 8 T + 94 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) | 4.79.aq_js_aecm_bumw |
| 83 | $D_4\times C_2$ | \( 1 - 224 T^{2} + 24702 T^{4} - 224 p^{2} T^{6} + p^{4} T^{8} \) | 4.83.a_aiq_a_bkoc |
| 89 | $C_4$ | \( ( 1 - 12 T + 134 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) | 4.89.ay_pw_ahxw_djxu |
| 97 | $D_4\times C_2$ | \( 1 - 60 T^{2} + 8198 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} \) | 4.97.a_aci_a_mdi |
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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
−6.14065012890214544075498420913, −5.77841019097279569951766684285, −5.77541680508138147545695572908, −5.40209469232089888119144544960, −5.38698138463593301482353761617, −5.26197679485636599262971335125, −4.94889687955864853474638722795, −4.86439944291757663856923202380, −4.46293562564894223021928260944, −4.28117497232112388484582144049, −3.94297449460738303786623934778, −3.80946902154871137871769186482, −3.80698210547081203460361208715, −3.48675076203734496056994922792, −3.13168796758480410601147706025, −2.75745384163787008883453174287, −2.73902428923425539813249218322, −2.33128151498723234822487987021, −2.10092858065140383020922882171, −2.07848791043544850350109552845, −2.06367487447976253667091362791, −1.07261792088207703195549534913, −0.873311138780902830700106636754, −0.814479557543222597392961862988, −0.23388674349230372837322854838,
0.23388674349230372837322854838, 0.814479557543222597392961862988, 0.873311138780902830700106636754, 1.07261792088207703195549534913, 2.06367487447976253667091362791, 2.07848791043544850350109552845, 2.10092858065140383020922882171, 2.33128151498723234822487987021, 2.73902428923425539813249218322, 2.75745384163787008883453174287, 3.13168796758480410601147706025, 3.48675076203734496056994922792, 3.80698210547081203460361208715, 3.80946902154871137871769186482, 3.94297449460738303786623934778, 4.28117497232112388484582144049, 4.46293562564894223021928260944, 4.86439944291757663856923202380, 4.94889687955864853474638722795, 5.26197679485636599262971335125, 5.38698138463593301482353761617, 5.40209469232089888119144544960, 5.77541680508138147545695572908, 5.77841019097279569951766684285, 6.14065012890214544075498420913
