| L(s) = 1 | − 4·3-s − 2·4-s + 4·9-s + 8·12-s − 5·16-s − 14·25-s + 4·27-s − 24·29-s − 8·36-s − 16·37-s + 4·43-s + 20·48-s − 24·49-s + 20·64-s + 36·73-s + 56·75-s − 12·79-s − 10·81-s − 36·83-s + 96·87-s + 24·89-s + 28·100-s − 4·103-s + 36·107-s − 8·108-s − 12·109-s + 64·111-s + ⋯ |
| L(s) = 1 | − 2.30·3-s − 4-s + 4/3·9-s + 2.30·12-s − 5/4·16-s − 2.79·25-s + 0.769·27-s − 4.45·29-s − 4/3·36-s − 2.63·37-s + 0.609·43-s + 2.88·48-s − 3.42·49-s + 5/2·64-s + 4.21·73-s + 6.46·75-s − 1.35·79-s − 1.11·81-s − 3.95·83-s + 10.2·87-s + 2.54·89-s + 14/5·100-s − 0.394·103-s + 3.48·107-s − 0.769·108-s − 1.14·109-s + 6.07·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(71^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(71^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | 71 | | \( 1 \) | |
| good | 2 | $C_2^2$ | \( ( 1 + T^{2} + p^{2} T^{4} )^{2} \) | 4.2.a_c_a_j |
| 3 | $D_{4}$ | \( ( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.3.e_m_bc_cg |
| 5 | $C_2^2$ | \( ( 1 + 7 T^{2} + p^{2} T^{4} )^{2} \) | 4.5.a_o_a_dv |
| 7 | $C_2^2$ | \( ( 1 + 12 T^{2} + p^{2} T^{4} )^{2} \) | 4.7.a_y_a_ji |
| 11 | $C_2^2$ | \( ( 1 + 16 T^{2} + p^{2} T^{4} )^{2} \) | 4.11.a_bg_a_te |
| 13 | $D_4\times C_2$ | \( 1 + 24 T^{2} + 290 T^{4} + 24 p^{2} T^{6} + p^{4} T^{8} \) | 4.13.a_y_a_le |
| 17 | $D_4\times C_2$ | \( 1 + 56 T^{2} + 1335 T^{4} + 56 p^{2} T^{6} + p^{4} T^{8} \) | 4.17.a_ce_a_bzj |
| 19 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) | 4.19.a_au_a_bfq |
| 23 | $D_4\times C_2$ | \( 1 + 44 T^{2} + 1110 T^{4} + 44 p^{2} T^{6} + p^{4} T^{8} \) | 4.23.a_bs_a_bqs |
| 29 | $D_{4}$ | \( ( 1 + 12 T + 82 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) | 4.29.y_lw_dym_yuo |
| 31 | $D_4\times C_2$ | \( 1 + 72 T^{2} + 3026 T^{4} + 72 p^{2} T^{6} + p^{4} T^{8} \) | 4.31.a_cu_a_emk |
| 37 | $D_{4}$ | \( ( 1 + 8 T + 63 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) | 4.37.q_hi_cjo_qyd |
| 41 | $D_4\times C_2$ | \( 1 + 80 T^{2} + 4287 T^{4} + 80 p^{2} T^{6} + p^{4} T^{8} \) | 4.41.a_dc_a_gix |
| 43 | $D_{4}$ | \( ( 1 - 2 T + 60 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.43.ae_eu_apw_lhy |
| 47 | $C_2^2$ | \( ( 1 + 88 T^{2} + p^{2} T^{4} )^{2} \) | 4.47.a_gu_a_rzu |
| 53 | $D_4\times C_2$ | \( 1 - 40 T^{2} + 3831 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \) | 4.53.a_abo_a_frj |
| 59 | $D_4\times C_2$ | \( 1 + 92 T^{2} + 5190 T^{4} + 92 p^{2} T^{6} + p^{4} T^{8} \) | 4.59.a_do_a_hrq |
| 61 | $D_4\times C_2$ | \( 1 - 24 T^{2} + 74 p T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} \) | 4.61.a_ay_a_grq |
| 67 | $D_4\times C_2$ | \( 1 + 24 T^{2} - 286 T^{4} + 24 p^{2} T^{6} + p^{4} T^{8} \) | 4.67.a_y_a_ala |
| 73 | $D_{4}$ | \( ( 1 - 18 T + 215 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \) | 4.73.abk_bda_apiu_fydf |
| 79 | $D_{4}$ | \( ( 1 + 6 T + 140 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.79.m_me_dxc_cdws |
| 83 | $D_{4}$ | \( ( 1 + 18 T + 220 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \) | 4.83.bk_bdk_qdo_gpoc |
| 89 | $D_{4}$ | \( ( 1 - 12 T + 166 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) | 4.89.ay_si_ajbk_dyda |
| 97 | $D_4\times C_2$ | \( 1 + 336 T^{2} + 46895 T^{4} + 336 p^{2} T^{6} + p^{4} T^{8} \) | 4.97.a_my_a_crjr |
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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
−6.09725944915774344596226430716, −5.87158815875206133737202797137, −5.75337993956611672363945406123, −5.54054880189030343166965754058, −5.53862741579005291989106481113, −5.15032151275469108525442841673, −5.10168965536269107894123163284, −5.06159381743958371629084528470, −4.87182080114437715324999657499, −4.39285418693747305986312580308, −4.18334896951715689448731671820, −4.12978609053251235091069924679, −4.07412443471595970815629955686, −3.55374923316788554573156023147, −3.46959854513726083154639482392, −3.35603075996931677896205658607, −3.26946421890960680072378971311, −2.65041482308950408914207860840, −2.39890676904683617662821549950, −2.00474164053669394757527680613, −1.98266035906666273753870420308, −1.83936349633638934290501543978, −1.58961222221748245419648393520, −1.07632222452990303290630401831, −0.65672565012218604415004526499, 0, 0, 0, 0,
0.65672565012218604415004526499, 1.07632222452990303290630401831, 1.58961222221748245419648393520, 1.83936349633638934290501543978, 1.98266035906666273753870420308, 2.00474164053669394757527680613, 2.39890676904683617662821549950, 2.65041482308950408914207860840, 3.26946421890960680072378971311, 3.35603075996931677896205658607, 3.46959854513726083154639482392, 3.55374923316788554573156023147, 4.07412443471595970815629955686, 4.12978609053251235091069924679, 4.18334896951715689448731671820, 4.39285418693747305986312580308, 4.87182080114437715324999657499, 5.06159381743958371629084528470, 5.10168965536269107894123163284, 5.15032151275469108525442841673, 5.53862741579005291989106481113, 5.54054880189030343166965754058, 5.75337993956611672363945406123, 5.87158815875206133737202797137, 6.09725944915774344596226430716
