| 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 + ⋯ |
| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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.i_cq_lk_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.ay_lg_adhw_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.ai_ka_acds_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
−6.90727219689775014162525202158, −6.83386254312843685526222133992, −6.63515393881984840420524307054, −6.03777946709549909170750957093, −5.86035809748366187555233794360, −5.63887906470805991706518211498, −5.61716972338139429731161729223, −5.39558164846142045528861163651, −5.25548343798241346493127484307, −4.91571240029925803441555444169, −4.75528904175595192242340437732, −4.62560460648512095216373261981, −4.05190285209349208127745644965, −3.75947497632796213646139191821, −3.72122957776554128220659963439, −3.45057419874259817876337104033, −2.84159582095365100080436358568, −2.82178546428874127849353441339, −2.77637623809755213216830777311, −2.38505760655077364117012964874, −2.21617798146916164013296282594, −1.31469913140239426573327330591, −1.10295173224343590109449848197, −1.08069263500393058636902178845, −0.40344577130341762491147327837,
0.40344577130341762491147327837, 1.08069263500393058636902178845, 1.10295173224343590109449848197, 1.31469913140239426573327330591, 2.21617798146916164013296282594, 2.38505760655077364117012964874, 2.77637623809755213216830777311, 2.82178546428874127849353441339, 2.84159582095365100080436358568, 3.45057419874259817876337104033, 3.72122957776554128220659963439, 3.75947497632796213646139191821, 4.05190285209349208127745644965, 4.62560460648512095216373261981, 4.75528904175595192242340437732, 4.91571240029925803441555444169, 5.25548343798241346493127484307, 5.39558164846142045528861163651, 5.61716972338139429731161729223, 5.63887906470805991706518211498, 5.86035809748366187555233794360, 6.03777946709549909170750957093, 6.63515393881984840420524307054, 6.83386254312843685526222133992, 6.90727219689775014162525202158
