| L(s) = 1 | + 3·9-s − 8·11-s + 8·19-s + 16·29-s − 16·31-s + 19·49-s + 14·59-s − 20·61-s − 20·71-s + 20·79-s − 7·81-s − 32·89-s − 24·99-s + 2·101-s − 12·109-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 39·169-s + 24·171-s + ⋯ |
| L(s) = 1 | + 9-s − 2.41·11-s + 1.83·19-s + 2.97·29-s − 2.87·31-s + 19/7·49-s + 1.82·59-s − 2.56·61-s − 2.37·71-s + 2.25·79-s − 7/9·81-s − 3.39·89-s − 2.41·99-s + 0.199·101-s − 1.14·109-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 + 3·169-s + 1.83·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.576456774\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.576456774\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{2} \) | |
| good | 3 | $D_4\times C_2$ | \( 1 - p T^{2} + 16 T^{4} - p^{3} T^{6} + p^{4} T^{8} \) | 4.3.a_ad_a_q |
| 7 | $D_4\times C_2$ | \( 1 - 19 T^{2} + 184 T^{4} - 19 p^{2} T^{6} + p^{4} T^{8} \) | 4.7.a_at_a_hc |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) | 4.11.i_cq_lk_bww |
| 13 | $D_4\times C_2$ | \( 1 - 3 p T^{2} + 680 T^{4} - 3 p^{3} T^{6} + p^{4} T^{8} \) | 4.13.a_abn_a_bae |
| 17 | $D_4\times C_2$ | \( 1 - 35 T^{2} + 676 T^{4} - 35 p^{2} T^{6} + p^{4} T^{8} \) | 4.17.a_abj_a_baa |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) | 4.19.ai_dw_asu_epa |
| 29 | $D_{4}$ | \( ( 1 - 8 T + 57 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) | 4.29.aq_gw_acay_mul |
| 31 | $D_{4}$ | \( ( 1 + 8 T + 61 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) | 4.31.q_he_ceq_ofr |
| 37 | $D_4\times C_2$ | \( 1 - 135 T^{2} + 7256 T^{4} - 135 p^{2} T^{6} + p^{4} T^{8} \) | 4.37.a_aff_a_ktc |
| 41 | $C_2^2$ | \( ( 1 + 65 T^{2} + p^{2} T^{4} )^{2} \) | 4.41.a_fa_a_lfv |
| 43 | $D_4\times C_2$ | \( 1 - 28 T^{2} + 2806 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} \) | 4.43.a_abc_a_edy |
| 47 | $D_4\times C_2$ | \( 1 - 99 T^{2} + 5912 T^{4} - 99 p^{2} T^{6} + p^{4} T^{8} \) | 4.47.a_adv_a_itk |
| 53 | $D_4\times C_2$ | \( 1 - 119 T^{2} + 8440 T^{4} - 119 p^{2} T^{6} + p^{4} T^{8} \) | 4.53.a_aep_a_mmq |
| 59 | $D_{4}$ | \( ( 1 - 7 T + 126 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) | 4.59.ao_lp_advq_bqiu |
| 61 | $D_{4}$ | \( ( 1 + 10 T + 130 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) | 4.61.u_nw_fqy_ccbm |
| 67 | $D_4\times C_2$ | \( 1 - 79 T^{2} + 9004 T^{4} - 79 p^{2} T^{6} + p^{4} T^{8} \) | 4.67.a_adb_a_nii |
| 71 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{4} \) | 4.71.u_qs_hbc_czep |
| 73 | $D_4\times C_2$ | \( 1 - 223 T^{2} + 22576 T^{4} - 223 p^{2} T^{6} + p^{4} T^{8} \) | 4.73.a_aip_a_bhki |
| 79 | $D_{4}$ | \( ( 1 - 10 T + 166 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) | 4.79.au_qq_ahgm_depq |
| 83 | $D_4\times C_2$ | \( 1 - 251 T^{2} + 29184 T^{4} - 251 p^{2} T^{6} + p^{4} T^{8} \) | 4.83.a_ajr_a_brem |
| 89 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) | 4.89.bg_bcm_prk_gvmk |
| 97 | $D_4\times C_2$ | \( 1 - 352 T^{2} + 49726 T^{4} - 352 p^{2} T^{6} + p^{4} T^{8} \) | 4.97.a_ano_a_cvoo |
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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
−5.67654031345287971081739404607, −5.60196377084548723833188083456, −5.51062878384414487217997523856, −5.18618767820284179086792746395, −5.07968298213947196156874154320, −5.07668122909022947842282295778, −4.86600892932828006148249790545, −4.32340726710853324088377595208, −4.28865254756555923735584879691, −4.18421908844144558034966174478, −3.96760058852112483535357257710, −3.59834244017124888450160823220, −3.53909129052607269127538042456, −3.01584720279808378849481351837, −2.93696195661227507510458966370, −2.85934757836434346491820283564, −2.62897703416786599333673922954, −2.38219157440588038329753007582, −2.15712172700942901618379956508, −1.71035421538282281554453408446, −1.53295005737256853242727538126, −1.16331532708904731999830937027, −1.10570844867588532389902393975, −0.55996791465514395342327538120, −0.19772924119624048150674476247,
0.19772924119624048150674476247, 0.55996791465514395342327538120, 1.10570844867588532389902393975, 1.16331532708904731999830937027, 1.53295005737256853242727538126, 1.71035421538282281554453408446, 2.15712172700942901618379956508, 2.38219157440588038329753007582, 2.62897703416786599333673922954, 2.85934757836434346491820283564, 2.93696195661227507510458966370, 3.01584720279808378849481351837, 3.53909129052607269127538042456, 3.59834244017124888450160823220, 3.96760058852112483535357257710, 4.18421908844144558034966174478, 4.28865254756555923735584879691, 4.32340726710853324088377595208, 4.86600892932828006148249790545, 5.07668122909022947842282295778, 5.07968298213947196156874154320, 5.18618767820284179086792746395, 5.51062878384414487217997523856, 5.60196377084548723833188083456, 5.67654031345287971081739404607
