| L(s) = 1 | + 4·3-s + 4·4-s + 6·9-s + 16·12-s + 12·16-s − 8·19-s − 4·25-s − 4·27-s + 24·36-s − 40·43-s + 48·48-s − 2·49-s − 32·57-s + 32·64-s + 24·67-s + 40·73-s − 16·75-s − 32·76-s − 37·81-s − 8·97-s − 16·100-s − 16·108-s − 20·121-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 2·4-s + 2·9-s + 4.61·12-s + 3·16-s − 1.83·19-s − 4/5·25-s − 0.769·27-s + 4·36-s − 6.09·43-s + 6.92·48-s − 2/7·49-s − 4.23·57-s + 4·64-s + 2.93·67-s + 4.68·73-s − 1.84·75-s − 3.67·76-s − 4.11·81-s − 0.812·97-s − 8/5·100-s − 1.53·108-s − 1.81·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.375441600\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.375441600\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) | |
| 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) | |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) | |
| good | 5 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) | 4.5.a_e_a_cc |
| 11 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) | 4.11.a_u_a_ne |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) | 4.13.a_au_a_qw |
| 17 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) | 4.17.a_aca_a_bwg |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) | 4.19.i_dw_su_epa |
| 23 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) | 4.23.a_cy_a_dsg |
| 29 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) | 4.29.a_ca_a_dms |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.31.a_aeu_a_inu |
| 37 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) | 4.37.a_au_a_efe |
| 41 | $C_2^2$ | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \) | 4.41.a_afs_a_nby |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) | 4.43.bo_bds_noi_edny |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.47.a_hg_a_tpu |
| 53 | $C_2^2$ | \( ( 1 + 74 T^{2} + p^{2} T^{4} )^{2} \) | 4.53.a_fs_a_qks |
| 59 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \) | 4.59.a_ado_a_nle |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \) | 4.61.a_bs_a_lsw |
| 67 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) | 4.67.ay_sq_aiku_dgpa |
| 71 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) | 4.71.a_fk_a_weg |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{4} \) | 4.73.abo_bii_aswu_hjrq |
| 79 | $C_2^2$ | \( ( 1 - 94 T^{2} + p^{2} T^{4} )^{2} \) | 4.79.a_ahg_a_bfny |
| 83 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2}( 1 + 18 T + p T^{2} )^{2} \) | 4.83.a_ame_a_cfic |
| 89 | $C_2^2$ | \( ( 1 - 170 T^{2} + p^{2} T^{4} )^{2} \) | 4.89.a_anc_a_coew |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) | 4.97.i_pw_dmu_dmla |
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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
−9.610473966394151717831103315116, −9.016118934977986001684017364111, −8.719073890077931542499335182637, −8.300822158962936545418024727970, −8.170281262010503247767184607189, −8.145325451573361101545997775255, −8.107744077875474269749494765804, −7.53128917723646651206989951814, −7.23945070790423035189528420544, −6.68528013093385233077056108291, −6.68414556374589557404038928983, −6.58899892600173745310000664358, −6.23960323117966823934363342095, −5.54540533642596049058520222448, −5.49382774102635178437791917358, −5.01905374252924603752083270309, −4.65224520937147804582866899607, −3.75457187543870857280883232826, −3.75276112875974796156846814344, −3.55523307825622633029163165586, −3.02547126240570724489787685404, −2.74296786811854716823813494487, −2.10625622678619304219159992543, −2.01262040497359072834218006410, −1.72582014390493601132796804977,
1.72582014390493601132796804977, 2.01262040497359072834218006410, 2.10625622678619304219159992543, 2.74296786811854716823813494487, 3.02547126240570724489787685404, 3.55523307825622633029163165586, 3.75276112875974796156846814344, 3.75457187543870857280883232826, 4.65224520937147804582866899607, 5.01905374252924603752083270309, 5.49382774102635178437791917358, 5.54540533642596049058520222448, 6.23960323117966823934363342095, 6.58899892600173745310000664358, 6.68414556374589557404038928983, 6.68528013093385233077056108291, 7.23945070790423035189528420544, 7.53128917723646651206989951814, 8.107744077875474269749494765804, 8.145325451573361101545997775255, 8.170281262010503247767184607189, 8.300822158962936545418024727970, 8.719073890077931542499335182637, 9.016118934977986001684017364111, 9.610473966394151717831103315116
