| L(s) = 1 | − 2·5-s + 6·9-s − 2·17-s + 11·25-s − 6·37-s − 32·41-s − 12·45-s − 2·49-s − 42·53-s − 42·61-s + 30·73-s + 27·81-s + 4·85-s + 26·89-s + 14·101-s − 10·109-s + 3·121-s − 38·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 12·153-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 2·9-s − 0.485·17-s + 11/5·25-s − 0.986·37-s − 4.99·41-s − 1.78·45-s − 2/7·49-s − 5.76·53-s − 5.37·61-s + 3.51·73-s + 3·81-s + 0.433·85-s + 2.75·89-s + 1.39·101-s − 0.957·109-s + 3/11·121-s − 3.39·125-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 − 0.970·153-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8413663275\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8413663275\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) | |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) | |
| good | 5 | $C_2^2$ | \( ( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) | 4.5.c_ah_c_cy |
| 11 | $C_2^3$ | \( 1 - 3 T^{2} - 112 T^{4} - 3 p^{2} T^{6} + p^{4} T^{8} \) | 4.11.a_ad_a_aei |
| 13 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) | 4.13.a_abc_a_uo |
| 17 | $C_2^2$ | \( ( 1 + T - 16 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) | 4.17.c_abf_c_bhk |
| 19 | $C_2^2$$\times$$C_2^2$ | \( ( 1 + 11 T^{2} + p^{2} T^{4} )( 1 + 26 T^{2} + p^{2} T^{4} ) \) | 4.19.a_bl_a_bmu |
| 23 | $C_2^3$ | \( 1 + 21 T^{2} - 88 T^{4} + 21 p^{2} T^{6} + p^{4} T^{8} \) | 4.23.a_v_a_adk |
| 29 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \) | 4.29.a_ado_a_fqc |
| 31 | $C_2^3$ | \( 1 - 19 T^{2} - 600 T^{4} - 19 p^{2} T^{6} + p^{4} T^{8} \) | 4.31.a_at_a_axc |
| 37 | $C_2^2$ | \( ( 1 + 3 T - 28 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) | 4.37.g_abv_cc_gfc |
| 41 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) | 4.41.bg_vc_iwe_cpoo |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.43.a_gq_a_qks |
| 47 | $C_2^3$ | \( 1 - 91 T^{2} + 6072 T^{4} - 91 p^{2} T^{6} + p^{4} T^{8} \) | 4.47.a_adn_a_izo |
| 53 | $C_2^2$ | \( ( 1 + 21 T + 200 T^{2} + 21 p T^{3} + p^{2} T^{4} )^{2} \) | 4.53.bq_bgj_pss_fgqm |
| 59 | $C_2^3$ | \( 1 - 43 T^{2} - 1632 T^{4} - 43 p^{2} T^{6} + p^{4} T^{8} \) | 4.59.a_abr_a_acku |
| 61 | $C_2^2$ | \( ( 1 + 21 T + 208 T^{2} + 21 p T^{3} + p^{2} T^{4} )^{2} \) | 4.61.bq_bgz_qso_fypo |
| 67 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 109 T^{2} + p^{2} T^{4} )( 1 + 122 T^{2} + p^{2} T^{4} ) \) | 4.67.a_n_a_agke |
| 71 | $C_2^2$ | \( ( 1 - 138 T^{2} + p^{2} T^{4} )^{2} \) | 4.71.a_akq_a_brcg |
| 73 | $C_2^2$ | \( ( 1 - 15 T + 148 T^{2} - 15 p T^{3} + p^{2} T^{4} )^{2} \) | 4.73.abe_ub_ajva_dstw |
| 79 | $C_2^3$ | \( 1 - 155 T^{2} + 17784 T^{4} - 155 p^{2} T^{6} + p^{4} T^{8} \) | 4.79.a_afz_a_baia |
| 83 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) | 4.83.a_aca_a_vjy |
| 89 | $C_2^2$ | \( ( 1 - 13 T + 80 T^{2} - 13 p T^{3} + p^{2} T^{4} )^{2} \) | 4.89.aba_mr_agna_czkm |
| 97 | $C_2^2$ | \( ( 1 - 182 T^{2} + p^{2} T^{4} )^{2} \) | 4.97.a_aoa_a_cyvu |
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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
−7.56567377567572924432667711568, −7.43726242189230491381760796311, −7.00052384979541171784723511517, −6.87374961567870506529979097335, −6.50465166870952610464025306998, −6.48493682941461070962935446647, −6.34062073996813680506728900141, −6.26058112726745435626685297025, −5.49302041441799309490506587364, −5.22888760136331035188951355902, −4.91471594233920015700779617363, −4.77290618329426056117389352783, −4.70267326941240199478200948030, −4.67484123865719220902953255915, −4.01733358191744714963154794214, −3.78835519489972937040244494181, −3.48121328455370505507698299020, −3.24760287114128486590638123730, −3.07842873938826525849999839885, −2.82274679960634504429462826564, −1.85001151088652100575298502003, −1.84395372839307911294820081728, −1.62154617051077850456520448401, −1.16921004664783307488677939992, −0.26123722547201962325188344290,
0.26123722547201962325188344290, 1.16921004664783307488677939992, 1.62154617051077850456520448401, 1.84395372839307911294820081728, 1.85001151088652100575298502003, 2.82274679960634504429462826564, 3.07842873938826525849999839885, 3.24760287114128486590638123730, 3.48121328455370505507698299020, 3.78835519489972937040244494181, 4.01733358191744714963154794214, 4.67484123865719220902953255915, 4.70267326941240199478200948030, 4.77290618329426056117389352783, 4.91471594233920015700779617363, 5.22888760136331035188951355902, 5.49302041441799309490506587364, 6.26058112726745435626685297025, 6.34062073996813680506728900141, 6.48493682941461070962935446647, 6.50465166870952610464025306998, 6.87374961567870506529979097335, 7.00052384979541171784723511517, 7.43726242189230491381760796311, 7.56567377567572924432667711568
