| L(s) = 1 | + 4·2-s + 10·4-s − 4·5-s + 4·7-s + 20·8-s − 6·9-s − 16·10-s − 8·11-s − 4·13-s + 16·14-s + 35·16-s − 4·17-s − 24·18-s − 8·19-s − 40·20-s − 32·22-s − 2·25-s − 16·26-s + 4·27-s + 40·28-s − 4·29-s + 56·32-s − 16·34-s − 16·35-s − 60·36-s + 8·37-s − 32·38-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 5·4-s − 1.78·5-s + 1.51·7-s + 7.07·8-s − 2·9-s − 5.05·10-s − 2.41·11-s − 1.10·13-s + 4.27·14-s + 35/4·16-s − 0.970·17-s − 5.65·18-s − 1.83·19-s − 8.94·20-s − 6.82·22-s − 2/5·25-s − 3.13·26-s + 0.769·27-s + 7.55·28-s − 0.742·29-s + 9.89·32-s − 2.74·34-s − 2.70·35-s − 10·36-s + 1.31·37-s − 5.19·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 7^{4} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 7^{4} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | 2 | $C_1$ | \( ( 1 - T )^{4} \) | |
| 7 | $C_1$ | \( ( 1 - T )^{4} \) | |
| 23 | | \( 1 \) | |
| good | 3 | $C_2 \wr C_2\wr C_2$ | \( 1 + 2 p T^{2} - 4 T^{3} + 20 T^{4} - 4 p T^{5} + 2 p^{3} T^{6} + p^{4} T^{8} \) | 4.3.a_g_ae_u |
| 5 | $C_2 \wr C_2\wr C_2$ | \( 1 + 4 T + 18 T^{2} + 8 p T^{3} + 116 T^{4} + 8 p^{2} T^{5} + 18 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.5.e_s_bo_em |
| 11 | $D_{4}$ | \( ( 1 + 4 T + 24 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.11.i_cm_ku_bta |
| 13 | $C_2 \wr C_2\wr C_2$ | \( 1 + 4 T + 40 T^{2} + 12 p T^{3} + 710 T^{4} + 12 p^{2} T^{5} + 40 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.13.e_bo_ga_bbi |
| 17 | $C_2 \wr C_2\wr C_2$ | \( 1 + 4 T + 66 T^{2} + 200 T^{3} + 1668 T^{4} + 200 p T^{5} + 66 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.17.e_co_hs_cme |
| 19 | $C_2 \wr C_2\wr C_2$ | \( 1 + 8 T + 4 p T^{2} + 424 T^{3} + 2182 T^{4} + 424 p T^{5} + 4 p^{3} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.19.i_cy_qi_dfy |
| 29 | $C_2 \wr C_2\wr C_2$ | \( 1 + 4 T + 60 T^{2} - 28 T^{3} + 1194 T^{4} - 28 p T^{5} + 60 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.29.e_ci_abc_bty |
| 31 | $C_2 \wr C_2\wr C_2$ | \( 1 + 86 T^{2} - 12 T^{3} + 3508 T^{4} - 12 p T^{5} + 86 p^{2} T^{6} + p^{4} T^{8} \) | 4.31.a_di_am_fey |
| 37 | $C_2 \wr C_2\wr C_2$ | \( 1 - 8 T + 128 T^{2} - 808 T^{3} + 6898 T^{4} - 808 p T^{5} + 128 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.37.ai_ey_abfc_kfi |
| 41 | $C_2 \wr C_2\wr C_2$ | \( 1 + 68 T^{2} + 256 T^{3} + 2102 T^{4} + 256 p T^{5} + 68 p^{2} T^{6} + p^{4} T^{8} \) | 4.41.a_cq_jw_dcw |
| 43 | $C_2 \wr C_2\wr C_2$ | \( 1 + 8 T + 68 T^{2} + 680 T^{3} + 4998 T^{4} + 680 p T^{5} + 68 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.43.i_cq_bae_hkg |
| 47 | $C_2 \wr C_2\wr C_2$ | \( 1 + 78 T^{2} + 532 T^{3} + 2228 T^{4} + 532 p T^{5} + 78 p^{2} T^{6} + p^{4} T^{8} \) | 4.47.a_da_um_dhs |
| 53 | $C_2^2 \wr C_2$ | \( 1 + 88 T^{2} + 7042 T^{4} + 88 p^{2} T^{6} + p^{4} T^{8} \) | 4.53.a_dk_a_kkw |
| 59 | $C_2 \wr C_2\wr C_2$ | \( 1 - 8 T + 222 T^{2} - 1284 T^{3} + 19172 T^{4} - 1284 p T^{5} + 222 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.59.ai_io_abxk_bcjk |
| 61 | $C_2 \wr C_2\wr C_2$ | \( 1 + 12 T + 186 T^{2} + 1488 T^{3} + 14484 T^{4} + 1488 p T^{5} + 186 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.61.m_he_cfg_vlc |
| 67 | $C_2 \wr C_2\wr C_2$ | \( 1 + 32 T + 608 T^{2} + 7712 T^{3} + 73458 T^{4} + 7712 p T^{5} + 608 p^{2} T^{6} + 32 p^{3} T^{7} + p^{4} T^{8} \) | 4.67.bg_xk_lkq_eeri |
| 71 | $C_2 \wr C_2\wr C_2$ | \( 1 - 8 T + 244 T^{2} - 1672 T^{3} + 24598 T^{4} - 1672 p T^{5} + 244 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.71.ai_jk_acmi_bkkc |
| 73 | $C_2 \wr C_2\wr C_2$ | \( 1 - 4 T + 72 T^{2} - 524 T^{3} + 5622 T^{4} - 524 p T^{5} + 72 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.73.ae_cu_aue_iig |
| 79 | $C_2 \wr C_2\wr C_2$ | \( 1 + 16 T + 128 T^{2} + 1008 T^{3} + 9570 T^{4} + 1008 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) | 4.79.q_ey_bmu_oec |
| 83 | $C_2 \wr C_2\wr C_2$ | \( 1 - 20 T + 464 T^{2} - 5332 T^{3} + 63574 T^{4} - 5332 p T^{5} + 464 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) | 4.83.au_rw_ahxc_dqbe |
| 89 | $C_2 \wr C_2\wr C_2$ | \( 1 + 28 T + 362 T^{2} + 2224 T^{3} + 12340 T^{4} + 2224 p T^{5} + 362 p^{2} T^{6} + 28 p^{3} T^{7} + p^{4} T^{8} \) | 4.89.bc_ny_dho_sgq |
| 97 | $C_2 \wr C_2\wr C_2$ | \( 1 - 16 T + 374 T^{2} - 4564 T^{3} + 53676 T^{4} - 4564 p T^{5} + 374 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) | 4.97.aq_ok_agto_dbkm |
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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
−6.00566205008428040034943094338, −5.48005293245871832477808506968, −5.44486653538318404106353659754, −5.30531234357535416933223110212, −5.16916285373074580343731506948, −4.88184375386966098741995743886, −4.72405544953181556043929181759, −4.72050045029346929099808465857, −4.49585540475849936862785230356, −4.20413295622050562162612035582, −4.15789382128462550855632306443, −3.92414006341933682041341136809, −3.76703877949356341181853191954, −3.48924412274563855909657875553, −3.33585979290668542498784171864, −3.08333701530155560523940338024, −2.77844689125762301277152475386, −2.52373087113946499360204248169, −2.52360299566093423014012907043, −2.38385625127464588037371435864, −2.38213332186183694826024832539, −1.83730996047445307291389016050, −1.69314216198930349475120799800, −1.23627389599137237567469465664, −1.10966862084587291299374390623, 0, 0, 0, 0,
1.10966862084587291299374390623, 1.23627389599137237567469465664, 1.69314216198930349475120799800, 1.83730996047445307291389016050, 2.38213332186183694826024832539, 2.38385625127464588037371435864, 2.52360299566093423014012907043, 2.52373087113946499360204248169, 2.77844689125762301277152475386, 3.08333701530155560523940338024, 3.33585979290668542498784171864, 3.48924412274563855909657875553, 3.76703877949356341181853191954, 3.92414006341933682041341136809, 4.15789382128462550855632306443, 4.20413295622050562162612035582, 4.49585540475849936862785230356, 4.72050045029346929099808465857, 4.72405544953181556043929181759, 4.88184375386966098741995743886, 5.16916285373074580343731506948, 5.30531234357535416933223110212, 5.44486653538318404106353659754, 5.48005293245871832477808506968, 6.00566205008428040034943094338
