| L(s) = 1 | − 4·4-s + 12·9-s − 4·11-s + 12·16-s + 10·25-s − 48·36-s + 16·44-s + 12·49-s − 56·59-s − 32·64-s + 90·81-s + 56·89-s − 48·99-s − 40·100-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 144·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 12·169-s + 173-s + ⋯ |
| L(s) = 1 | − 2·4-s + 4·9-s − 1.20·11-s + 3·16-s + 2·25-s − 8·36-s + 2.41·44-s + 12/7·49-s − 7.29·59-s − 4·64-s + 10·81-s + 5.93·89-s − 4.82·99-s − 4·100-s − 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 12·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.923·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.972441551\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.972441551\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) | |
| 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) | |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) | |
| good | 3 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.3.a_am_a_cc |
| 7 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) | 4.7.a_am_a_fe |
| 13 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) | 4.13.a_m_a_ok |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.17.a_cq_a_cos |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) | 4.19.a_e_a_bby |
| 23 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) | 4.23.a_ca_a_cos |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.29.a_em_a_hmc |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.31.a_aeu_a_inu |
| 37 | $C_2^2$ | \( ( 1 + 54 T^{2} + p^{2} T^{4} )^{2} \) | 4.37.a_ee_a_ijm |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2}( 1 + 2 T + p T^{2} )^{2} \) | 4.41.a_ga_a_nzi |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.43.a_gq_a_qks |
| 47 | $C_2^2$ | \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \) | 4.47.a_agq_a_rmk |
| 53 | $C_2^2$ | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \) | 4.53.a_afs_a_qks |
| 59 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{4} \) | 4.59.ce_cci_bexk_lhac |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.61.a_jk_a_bhas |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.67.a_aki_a_bnvy |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.71.a_aky_a_bsti |
| 73 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.73.a_lg_a_bvhu |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.79.a_me_a_cdkg |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.83.a_mu_a_cjdu |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{4} \) | 4.89.ace_cgy_abmjg_quuo |
| 97 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.97.a_aoy_a_dfni |
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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
−7.992730425508923991598868519625, −7.80744014108965148196020815365, −7.64240554973917845526523670754, −7.40555696673047135098283006278, −7.24139197262858442220818883544, −6.93351916125712065723752529059, −6.44041219922474905358417295256, −6.31989755884510459940441327433, −6.25296731197510701309726620582, −5.74571812027207175272375008306, −5.14425985890785610258325349282, −5.08216413510064223558623240462, −5.04431730783932112035103422947, −4.54997320584717936987077508143, −4.46035620326271424880492590244, −4.26034266005358671722942550034, −4.10972218283867298371812207538, −3.41197182418610706030178095552, −3.37271698274968351494439178054, −3.10816101653539683548709820226, −2.45334166700548109117744999400, −1.81635683300536364190647610937, −1.58710134003318877082264685722, −1.09323970824700329285792452623, −0.65108914477940139799924322261,
0.65108914477940139799924322261, 1.09323970824700329285792452623, 1.58710134003318877082264685722, 1.81635683300536364190647610937, 2.45334166700548109117744999400, 3.10816101653539683548709820226, 3.37271698274968351494439178054, 3.41197182418610706030178095552, 4.10972218283867298371812207538, 4.26034266005358671722942550034, 4.46035620326271424880492590244, 4.54997320584717936987077508143, 5.04431730783932112035103422947, 5.08216413510064223558623240462, 5.14425985890785610258325349282, 5.74571812027207175272375008306, 6.25296731197510701309726620582, 6.31989755884510459940441327433, 6.44041219922474905358417295256, 6.93351916125712065723752529059, 7.24139197262858442220818883544, 7.40555696673047135098283006278, 7.64240554973917845526523670754, 7.80744014108965148196020815365, 7.992730425508923991598868519625
