| L(s) = 1 | + 6·7-s + 6·11-s − 4·13-s + 6·19-s − 6·23-s + 14·25-s − 6·29-s − 12·41-s − 2·43-s + 16·49-s − 12·53-s − 48·59-s − 20·61-s + 6·67-s + 18·71-s + 36·77-s + 8·79-s + 12·89-s − 24·91-s − 18·101-s + 4·103-s − 6·107-s + 12·113-s + 8·121-s + 127-s + 131-s + 36·133-s + ⋯ |
| L(s) = 1 | + 2.26·7-s + 1.80·11-s − 1.10·13-s + 1.37·19-s − 1.25·23-s + 14/5·25-s − 1.11·29-s − 1.87·41-s − 0.304·43-s + 16/7·49-s − 1.64·53-s − 6.24·59-s − 2.56·61-s + 0.733·67-s + 2.13·71-s + 4.10·77-s + 0.900·79-s + 1.27·89-s − 2.51·91-s − 1.79·101-s + 0.394·103-s − 0.580·107-s + 1.12·113-s + 8/11·121-s + 0.0887·127-s + 0.0873·131-s + 3.12·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.620295362\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.620295362\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 3 | | \( 1 \) | |
| 13 | $C_2^2$ | \( 1 + 4 T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) | |
| good | 5 | $C_2^2$ | \( ( 1 - 7 T^{2} + p^{2} T^{4} )^{2} \) | 4.5.a_ao_a_dv |
| 7 | $D_4\times C_2$ | \( 1 - 6 T + 20 T^{2} - 48 T^{3} + 99 T^{4} - 48 p T^{5} + 20 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.7.ag_u_abw_dv |
| 11 | $D_4\times C_2$ | \( 1 - 6 T + 28 T^{2} - 96 T^{3} + 267 T^{4} - 96 p T^{5} + 28 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.11.ag_bc_ads_kh |
| 17 | $C_2^3$ | \( 1 - 7 T^{2} - 240 T^{4} - 7 p^{2} T^{6} + p^{4} T^{8} \) | 4.17.a_ah_a_ajg |
| 19 | $D_4\times C_2$ | \( 1 - 6 T + 44 T^{2} - 192 T^{3} + 891 T^{4} - 192 p T^{5} + 44 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.19.ag_bs_ahk_bih |
| 23 | $D_4\times C_2$ | \( 1 + 6 T + 8 T^{2} - 108 T^{3} - 573 T^{4} - 108 p T^{5} + 8 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.23.g_i_aee_awb |
| 29 | $C_2^2$ | \( ( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) | 4.29.g_abf_cc_dwe |
| 31 | $D_4\times C_2$ | \( 1 - 28 T^{2} + 390 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} \) | 4.31.a_abc_a_pa |
| 37 | $C_2^3$ | \( 1 + 65 T^{2} + 2856 T^{4} + 65 p^{2} T^{6} + p^{4} T^{8} \) | 4.37.a_cn_a_efw |
| 41 | $D_4\times C_2$ | \( 1 + 12 T + 133 T^{2} + 1020 T^{3} + 7512 T^{4} + 1020 p T^{5} + 133 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.41.m_fd_bng_lcy |
| 43 | $D_4\times C_2$ | \( 1 + 2 T - 56 T^{2} - 52 T^{3} + 1579 T^{4} - 52 p T^{5} - 56 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.43.c_ace_aca_cit |
| 47 | $D_4\times C_2$ | \( 1 - 164 T^{2} + 11034 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} \) | 4.47.a_agi_a_qik |
| 53 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) | 4.53.m_kg_czo_bhnn |
| 59 | $C_2^2$ | \( ( 1 + 24 T + 251 T^{2} + 24 p T^{3} + p^{2} T^{4} )^{2} \) | 4.59.bw_bpm_wai_hwbb |
| 61 | $D_4\times C_2$ | \( 1 + 20 T + 205 T^{2} + 1460 T^{3} + 9904 T^{4} + 1460 p T^{5} + 205 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) | 4.61.u_hx_cee_oqy |
| 67 | $D_4\times C_2$ | \( 1 - 6 T + 68 T^{2} - 336 T^{3} - 549 T^{4} - 336 p T^{5} + 68 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.67.ag_cq_amy_avd |
| 71 | $D_4\times C_2$ | \( 1 - 18 T + 268 T^{2} - 2880 T^{3} + 28227 T^{4} - 2880 p T^{5} + 268 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) | 4.71.as_ki_aegu_bptr |
| 73 | $C_2^2$ | \( ( 1 + T^{2} + p^{2} T^{4} )^{2} \) | 4.73.a_c_a_ptz |
| 79 | $D_{4}$ | \( ( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.79.ai_eu_aboy_banm |
| 83 | $D_4\times C_2$ | \( 1 - 164 T^{2} + 17802 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} \) | 4.83.a_agi_a_bais |
| 89 | $D_4\times C_2$ | \( 1 - 12 T + 202 T^{2} - 1848 T^{3} + 20067 T^{4} - 1848 p T^{5} + 202 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.89.am_hu_actc_bdrv |
| 97 | $C_2^3$ | \( 1 + 158 T^{2} + 15555 T^{4} + 158 p^{2} T^{6} + p^{4} T^{8} \) | 4.97.a_gc_a_xah |
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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
−6.58063965520517497160131735226, −6.31438649698125461466528407918, −6.30555865543827781469676838759, −5.81832308783579149623299073937, −5.73030876579598222204765510687, −5.54317749337125290890130895673, −5.06645029625215976865372083531, −5.01423448687350113837063226140, −4.74166693642063312789743566316, −4.73715921885503603109541029974, −4.52499841700204365896305674105, −4.37115210192223628964950792135, −3.99626069944786843634648815678, −3.58667196613795002298033152564, −3.45321071656689300032808078600, −3.16541067873596680834893929007, −2.94572041706913603725112969761, −2.90123078769148414230675656160, −2.17082205294584385439930047932, −1.98283609343866307418158810511, −1.66474614956849536468813274985, −1.60942909044485499508453933617, −1.28583531835747135660186836872, −0.970104190587649207750195131867, −0.30742993662083723638079682371,
0.30742993662083723638079682371, 0.970104190587649207750195131867, 1.28583531835747135660186836872, 1.60942909044485499508453933617, 1.66474614956849536468813274985, 1.98283609343866307418158810511, 2.17082205294584385439930047932, 2.90123078769148414230675656160, 2.94572041706913603725112969761, 3.16541067873596680834893929007, 3.45321071656689300032808078600, 3.58667196613795002298033152564, 3.99626069944786843634648815678, 4.37115210192223628964950792135, 4.52499841700204365896305674105, 4.73715921885503603109541029974, 4.74166693642063312789743566316, 5.01423448687350113837063226140, 5.06645029625215976865372083531, 5.54317749337125290890130895673, 5.73030876579598222204765510687, 5.81832308783579149623299073937, 6.30555865543827781469676838759, 6.31438649698125461466528407918, 6.58063965520517497160131735226
