| L(s) = 1 | + 2·4-s − 5·16-s − 8·19-s + 8·31-s + 20·49-s − 40·61-s − 20·64-s − 16·76-s + 16·79-s + 40·109-s − 20·121-s + 16·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 4-s − 5/4·16-s − 1.83·19-s + 1.43·31-s + 20/7·49-s − 5.12·61-s − 5/2·64-s − 1.83·76-s + 1.80·79-s + 3.83·109-s − 1.81·121-s + 1.43·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.153·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.737514230\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.737514230\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | | \( 1 \) | |
| 5 | | \( 1 \) | |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{2} \) | |
| good | 2 | $C_2^2$ | \( ( 1 - T^{2} + p^{2} T^{4} )^{2} \) | 4.2.a_ac_a_j |
| 7 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) | 4.7.a_au_a_hq |
| 11 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) | 4.11.a_u_a_ne |
| 17 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) | 4.17.a_bc_a_bdu |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) | 4.19.i_dw_su_epa |
| 23 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) | 4.23.a_e_a_bow |
| 29 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) | 4.29.a_u_a_cqo |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) | 4.31.ai_fs_abdw_ktq |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) | 4.37.a_afk_a_lhu |
| 41 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) | 4.41.a_cq_a_gru |
| 43 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) | 4.43.a_abs_a_gew |
| 47 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) | 4.47.a_bc_a_gvm |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.53.a_aie_a_yyg |
| 59 | $C_2^2$ | \( ( 1 + 106 T^{2} + p^{2} T^{4} )^{2} \) | 4.59.a_ie_a_baxy |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) | 4.61.bo_bgm_qtk_gacs |
| 67 | $C_2^2$ | \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2} \) | 4.67.a_eu_a_sze |
| 71 | $C_2^2$ | \( ( 1 + 130 T^{2} + p^{2} T^{4} )^{2} \) | 4.71.a_ka_a_bnxu |
| 73 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \) | 4.73.a_ado_a_sxi |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) | 4.79.aq_pw_afzs_dafm |
| 83 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) | 4.83.a_aem_a_zji |
| 89 | $C_2^2$ | \( ( 1 + 130 T^{2} + p^{2} T^{4} )^{2} \) | 4.89.a_ka_a_bwli |
| 97 | $C_2^2$ | \( ( 1 - 94 T^{2} + p^{2} T^{4} )^{2} \) | 4.97.a_ahg_a_boxq |
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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
−6.33667521123317372985277593092, −6.11617910835117565393654758465, −5.74710096825923939908363690228, −5.67804491588762768491590946936, −5.55759045154070788979653439559, −5.24658863438574082178813398988, −4.79307103875145028552725567594, −4.58582551978435087327153162889, −4.55106774943340329366770566473, −4.45435267427318885017037057862, −4.31311113416572596671201352315, −3.88969032024867705437802038694, −3.74627089746051257141118827603, −3.28989431015129472861471038317, −3.13777105191266907660845842579, −2.96669207862143173750172345935, −2.79185469229515332739538860600, −2.28625621297087126292755738529, −2.14719474997115338030361967289, −2.08159230916042271230957987179, −1.93990995362286380185102193009, −1.27491567267923539406844613668, −1.20731590201477604696342268204, −0.64296612699810335872346788223, −0.21417575453941567693903686781,
0.21417575453941567693903686781, 0.64296612699810335872346788223, 1.20731590201477604696342268204, 1.27491567267923539406844613668, 1.93990995362286380185102193009, 2.08159230916042271230957987179, 2.14719474997115338030361967289, 2.28625621297087126292755738529, 2.79185469229515332739538860600, 2.96669207862143173750172345935, 3.13777105191266907660845842579, 3.28989431015129472861471038317, 3.74627089746051257141118827603, 3.88969032024867705437802038694, 4.31311113416572596671201352315, 4.45435267427318885017037057862, 4.55106774943340329366770566473, 4.58582551978435087327153162889, 4.79307103875145028552725567594, 5.24658863438574082178813398988, 5.55759045154070788979653439559, 5.67804491588762768491590946936, 5.74710096825923939908363690228, 6.11617910835117565393654758465, 6.33667521123317372985277593092
