| L(s) = 1 | + 4-s + 2·7-s + 18·13-s − 12·19-s + 4·25-s + 2·28-s − 30·31-s + 10·37-s − 2·43-s − 11·49-s + 18·52-s + 6·61-s − 64-s + 26·67-s − 24·73-s − 12·76-s + 14·79-s + 36·91-s + 6·97-s + 4·100-s + 22·109-s + 44·121-s − 30·124-s + 127-s + 131-s − 24·133-s + 137-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 0.755·7-s + 4.99·13-s − 2.75·19-s + 4/5·25-s + 0.377·28-s − 5.38·31-s + 1.64·37-s − 0.304·43-s − 1.57·49-s + 2.49·52-s + 0.768·61-s − 1/8·64-s + 3.17·67-s − 2.80·73-s − 1.37·76-s + 1.57·79-s + 3.77·91-s + 0.609·97-s + 2/5·100-s + 2.10·109-s + 4·121-s − 2.69·124-s + 0.0887·127-s + 0.0873·131-s − 2.08·133-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.234080266\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.234080266\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) | |
| 3 | | \( 1 \) | |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) | |
| good | 5 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) | 4.5.a_ae_a_cc |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.11.a_abs_a_bby |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2}( 1 - 2 T + p T^{2} )^{2} \) | 4.13.as_gf_abks_fzo |
| 17 | $C_2^3$ | \( 1 + 14 T^{2} - 93 T^{4} + 14 p^{2} T^{6} + p^{4} T^{8} \) | 4.17.a_o_a_adp |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) | 4.19.m_du_xc_enj |
| 23 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) | 4.23.a_au_a_bso |
| 29 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) | 4.29.a_cg_a_dtb |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2}( 1 + 11 T + p T^{2} )^{2} \) | 4.31.be_qv_gcc_bocq |
| 37 | $C_2^2$ | \( ( 1 - 5 T - 12 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) | 4.37.ak_b_ajq_haa |
| 41 | $C_2^3$ | \( 1 - 34 T^{2} - 525 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \) | 4.41.a_abi_a_auf |
| 43 | $C_2^2$ | \( ( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) | 4.43.c_adf_c_ifk |
| 47 | $C_2^3$ | \( 1 - 46 T^{2} - 93 T^{4} - 46 p^{2} T^{6} + p^{4} T^{8} \) | 4.47.a_abu_a_adp |
| 53 | $C_2^3$ | \( 1 + 70 T^{2} + 2091 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8} \) | 4.53.a_cs_a_dcl |
| 59 | $C_2^3$ | \( 1 - 70 T^{2} + 1419 T^{4} - 70 p^{2} T^{6} + p^{4} T^{8} \) | 4.59.a_acs_a_ccp |
| 61 | $C_2^2$ | \( ( 1 - 3 T + 64 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) | 4.61.ag_fh_abcw_ssa |
| 67 | $C_2^2$ | \( ( 1 - 13 T + 102 T^{2} - 13 p T^{3} + p^{2} T^{4} )^{2} \) | 4.67.aba_oj_agna_ckem |
| 71 | $C_2^2$ | \( ( 1 - 106 T^{2} + p^{2} T^{4} )^{2} \) | 4.71.a_aie_a_bfny |
| 73 | $C_2^2$ | \( ( 1 + 12 T + 121 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) | 4.73.y_ow_gxc_cqnr |
| 79 | $C_2^2$ | \( ( 1 - 7 T - 30 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) | 4.79.ao_al_abak_bfgm |
| 83 | $C_2^3$ | \( 1 - 154 T^{2} + 16827 T^{4} - 154 p^{2} T^{6} + p^{4} T^{8} \) | 4.83.a_afy_a_yxf |
| 89 | $C_2^3$ | \( 1 - 70 T^{2} - 3021 T^{4} - 70 p^{2} T^{6} + p^{4} T^{8} \) | 4.89.a_acs_a_aemf |
| 97 | $C_2^2$ | \( ( 1 - 3 T + 100 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) | 4.97.ag_ib_abtm_btfo |
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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
−6.96378026123602187245090680843, −6.83115991368779958834534872470, −6.51955314544245385921659048027, −6.28899952274190684794586431328, −6.07144734353037755321345358304, −6.00457091086607445018291338383, −5.82382758513697999875299612455, −5.53677288571378977613681156721, −5.36135529618011675339660585359, −5.00546586301644104064392790759, −4.64152600345882242733972865123, −4.41552384747140738778809795764, −4.24466287493884740053332112215, −3.85162777668728726353291850999, −3.62117535414110725256865446361, −3.61852164188572521199129828852, −3.30555717565926101217848575539, −3.23016112147725485361010732927, −2.34541692072223609313900296505, −2.24218084898688641667300516939, −2.00089476286532776473425344788, −1.57988232738071881661164918240, −1.44812717591421672896527115259, −1.06365222997550164639941068247, −0.43116114068380460526076196448,
0.43116114068380460526076196448, 1.06365222997550164639941068247, 1.44812717591421672896527115259, 1.57988232738071881661164918240, 2.00089476286532776473425344788, 2.24218084898688641667300516939, 2.34541692072223609313900296505, 3.23016112147725485361010732927, 3.30555717565926101217848575539, 3.61852164188572521199129828852, 3.62117535414110725256865446361, 3.85162777668728726353291850999, 4.24466287493884740053332112215, 4.41552384747140738778809795764, 4.64152600345882242733972865123, 5.00546586301644104064392790759, 5.36135529618011675339660585359, 5.53677288571378977613681156721, 5.82382758513697999875299612455, 6.00457091086607445018291338383, 6.07144734353037755321345358304, 6.28899952274190684794586431328, 6.51955314544245385921659048027, 6.83115991368779958834534872470, 6.96378026123602187245090680843
