| L(s) = 1 | − 12·9-s + 10·25-s + 8·41-s − 12·49-s + 90·81-s − 56·89-s + 36·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 12·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s − 120·225-s + ⋯ |
| L(s) = 1 | − 4·9-s + 2·25-s + 1.24·41-s − 1.71·49-s + 10·81-s − 5.93·89-s + 3.27·121-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 − 0.923·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s − 8·225-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6780267831\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6780267831\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) | |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.3.a_m_a_cc |
| 7 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) | 4.7.a_m_a_fe |
| 11 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) | 4.11.a_abk_a_vu |
| 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_acq_a_cos |
| 19 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) | 4.19.a_ae_a_bby |
| 23 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) | 4.23.a_aca_a_cos |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.29.a_aem_a_hmc |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.31.a_eu_a_inu |
| 37 | $C_2^2$ | \( ( 1 + 54 T^{2} + p^{2} T^{4} )^{2} \) | 4.37.a_ee_a_ijm |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) | 4.41.ai_hg_abnc_rwg |
| 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_gq_a_rmk |
| 53 | $C_2^2$ | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \) | 4.53.a_afs_a_qks |
| 59 | $C_2^2$ | \( ( 1 + 78 T^{2} + p^{2} T^{4} )^{2} \) | 4.59.a_ga_a_thu |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.61.a_ajk_a_bhas |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.67.a_ki_a_bnvy |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.71.a_ky_a_bsti |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.73.a_alg_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.ce_cgy_bmjg_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
−8.395522189463721427432451945155, −8.273235517327712159386382596749, −8.271948384891855967724027729889, −7.79146860430222018772902531898, −7.36117283476782268155519649770, −7.34280393125255434375117540885, −6.78773385258931231824529573579, −6.60472382004018630833946462459, −6.40560275103942236216451918338, −5.86913560571097498187262524198, −5.83546784057651874510305314798, −5.79347639874416465994420150139, −5.22023496169602041334162095360, −5.07115260460609805077004276843, −4.93953054081686779578491434614, −4.29530307611454374512250789960, −4.17784388377032040855948361174, −3.56692841494322765161433522685, −3.10570645361687311255346272720, −3.06608783253236899607480695920, −2.83092772990423054658736886172, −2.45746584430386316812710546766, −2.01939718674622824220878174677, −1.20604689492161767630898106379, −0.38329903122714331145618322510,
0.38329903122714331145618322510, 1.20604689492161767630898106379, 2.01939718674622824220878174677, 2.45746584430386316812710546766, 2.83092772990423054658736886172, 3.06608783253236899607480695920, 3.10570645361687311255346272720, 3.56692841494322765161433522685, 4.17784388377032040855948361174, 4.29530307611454374512250789960, 4.93953054081686779578491434614, 5.07115260460609805077004276843, 5.22023496169602041334162095360, 5.79347639874416465994420150139, 5.83546784057651874510305314798, 5.86913560571097498187262524198, 6.40560275103942236216451918338, 6.60472382004018630833946462459, 6.78773385258931231824529573579, 7.34280393125255434375117540885, 7.36117283476782268155519649770, 7.79146860430222018772902531898, 8.271948384891855967724027729889, 8.273235517327712159386382596749, 8.395522189463721427432451945155
