| L(s) = 1 | + 4·9-s + 20·25-s − 28·49-s − 6·81-s − 72·89-s − 40·97-s − 72·113-s − 14·121-s + ⋯ |
| L(s) = 1 | + 4/3·9-s + 4·25-s − 4·49-s − 2/3·81-s − 7.63·89-s − 4.06·97-s − 6.77·113-s − 1.27·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4574879783\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4574879783\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) | |
| good | 3 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) | 4.3.a_ae_a_w |
| 5 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.5.a_au_a_fu |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.7.a_bc_a_li |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.13.a_ca_a_bna |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) | 4.17.a_ae_a_wk |
| 19 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) | 4.19.a_acq_a_cug |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.23.a_ado_a_esc |
| 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 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.37.a_afs_a_mdy |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) | 4.41.a_do_a_ics |
| 43 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) | 4.43.a_bc_a_ftu |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.47.a_ahg_a_tpu |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.53.a_aie_a_yyg |
| 59 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) | 4.59.a_agi_a_ugk |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.61.a_jk_a_bhas |
| 67 | $C_2^2$ | \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2} \) | 4.67.a_eu_a_sze |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.71.a_aky_a_bsti |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2}( 1 + 2 T + p T^{2} )^{2} \) | 4.73.a_ky_a_btpm |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.79.a_me_a_cdkg |
| 83 | $C_2^2$ | \( ( 1 + 158 T^{2} + p^{2} T^{4} )^{2} \) | 4.83.a_me_a_cfic |
| 89 | $C_2$ | \( ( 1 + 18 T + p T^{2} )^{4} \) | 4.89.cu_dkm_ckyq_bcjmk |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) | 4.97.bo_bma_xdo_kkmw |
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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
−6.68639453885800202982140432643, −6.65558122589162475371907278853, −6.48280702931901403855735952025, −6.42582662587775453550679157224, −5.98522898270704782586369749591, −5.64846287837825106954886516346, −5.35399014448285097058116714963, −5.14716656469764117334284288157, −5.14252769636614615656199984058, −5.00085047111000593626639789701, −4.34969761912690698626177881418, −4.34268895522698667531425595810, −4.18297576374913806686871189126, −4.11365017378342522745373497110, −3.61707659447007285347013467205, −3.11698561832902122456914914289, −3.11472986555328318776700767475, −2.81018047411456554670468211702, −2.65797077235120219091571666030, −2.41124701856803431607961119967, −1.60936960349734452831270489327, −1.44637648437853610515470390283, −1.30036754980955717982717452346, −1.17478332145493555710386718957, −0.12392439755203290766652259253,
0.12392439755203290766652259253, 1.17478332145493555710386718957, 1.30036754980955717982717452346, 1.44637648437853610515470390283, 1.60936960349734452831270489327, 2.41124701856803431607961119967, 2.65797077235120219091571666030, 2.81018047411456554670468211702, 3.11472986555328318776700767475, 3.11698561832902122456914914289, 3.61707659447007285347013467205, 4.11365017378342522745373497110, 4.18297576374913806686871189126, 4.34268895522698667531425595810, 4.34969761912690698626177881418, 5.00085047111000593626639789701, 5.14252769636614615656199984058, 5.14716656469764117334284288157, 5.35399014448285097058116714963, 5.64846287837825106954886516346, 5.98522898270704782586369749591, 6.42582662587775453550679157224, 6.48280702931901403855735952025, 6.65558122589162475371907278853, 6.68639453885800202982140432643
