| L(s) = 1 | − 3·5-s + 5·9-s + 4·11-s + 16·19-s + 5·25-s − 12·29-s − 2·31-s + 12·41-s − 15·45-s + 4·49-s − 12·55-s − 18·59-s + 20·61-s + 6·71-s + 28·79-s + 9·81-s + 6·89-s − 48·95-s + 20·99-s + 24·101-s − 40·109-s + 10·121-s − 18·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 5/3·9-s + 1.20·11-s + 3.67·19-s + 25-s − 2.22·29-s − 0.359·31-s + 1.87·41-s − 2.23·45-s + 4/7·49-s − 1.61·55-s − 2.34·59-s + 2.56·61-s + 0.712·71-s + 3.15·79-s + 81-s + 0.635·89-s − 4.92·95-s + 2.01·99-s + 2.38·101-s − 3.83·109-s + 0.909·121-s − 1.60·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.069559505\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.069559505\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | |
| 11 | $C_1$ | \( ( 1 - T )^{4} \) | |
| good | 3 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} )( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} ) \) | 4.3.a_af_a_q |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) | 4.7.a_ae_a_dy |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.13.a_aca_a_bna |
| 17 | $D_4\times C_2$ | \( 1 - 40 T^{2} + 846 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \) | 4.17.a_abo_a_bgo |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) | 4.19.aq_gq_absy_izm |
| 23 | $D_4\times C_2$ | \( 1 - 85 T^{2} + 2856 T^{4} - 85 p^{2} T^{6} + p^{4} T^{8} \) | 4.23.a_adh_a_efw |
| 29 | $D_{4}$ | \( ( 1 + 6 T + 34 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.29.m_ea_bdc_hhm |
| 31 | $D_{4}$ | \( ( 1 + T + 54 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) | 4.31.c_ef_go_hgm |
| 37 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 7 T + 12 T^{2} - 7 p T^{3} + p^{2} T^{4} )( 1 + 7 T + 12 T^{2} + 7 p T^{3} + p^{2} T^{4} ) \) | 4.37.a_az_a_abcq |
| 41 | $D_{4}$ | \( ( 1 - 6 T + 58 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.41.am_fw_abts_oig |
| 43 | $C_2^2$ | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \) | 4.43.a_afs_a_now |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) | 4.47.a_adw_a_kgc |
| 53 | $D_4\times C_2$ | \( 1 - 100 T^{2} + 6006 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \) | 4.53.a_adw_a_ixa |
| 59 | $D_{4}$ | \( ( 1 + 9 T + 130 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) | 4.59.s_nd_faw_bxlk |
| 61 | $D_{4}$ | \( ( 1 - 10 T + 114 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) | 4.61.au_mq_afeq_bwhi |
| 67 | $D_4\times C_2$ | \( 1 - 181 T^{2} + 15312 T^{4} - 181 p^{2} T^{6} + p^{4} T^{8} \) | 4.67.a_agz_a_wqy |
| 71 | $D_{4}$ | \( ( 1 - 3 T + 70 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) | 4.71.ag_ft_abgo_ybk |
| 73 | $C_2^2$ | \( ( 1 - 98 T^{2} + p^{2} T^{4} )^{2} \) | 4.73.a_aho_a_bdzi |
| 79 | $D_{4}$ | \( ( 1 - 14 T + 174 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) | 4.79.abc_uy_akmm_efbq |
| 83 | $C_2^2$ | \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{2} \) | 4.83.a_ajk_a_bqkk |
| 89 | $D_{4}$ | \( ( 1 - 3 T + 172 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) | 4.89.ag_np_acig_crou |
| 97 | $D_4\times C_2$ | \( 1 - 337 T^{2} + 47136 T^{4} - 337 p^{2} T^{6} + p^{4} T^{8} \) | 4.97.a_amz_a_crsy |
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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
−7.48250272987207871630447717625, −7.01064618409550120197734454941, −6.90666079725285702805107729196, −6.70984283365105840440582208021, −6.53818820367562328796983964398, −6.13258299604999053346447391076, −5.93548737642554884987300680562, −5.48123434591653625900793415515, −5.33353134253557802817719305476, −5.27603629498225512980487398888, −5.00840114273274858489610625466, −4.52081601147585233230808931139, −4.41534783016159892132341730645, −3.97798413278449709741089876212, −3.93611836399217042815123225186, −3.68674720568590558888193442510, −3.54663854481281249684859923722, −3.09593886255877725277486805576, −2.96343353590840836911262861828, −2.49998602546216877371964709139, −1.87971754577122038881323764731, −1.80244748863381436360324815927, −1.26853743653221450721012498261, −0.881509004708087573268278527555, −0.66184193945979415666324925277,
0.66184193945979415666324925277, 0.881509004708087573268278527555, 1.26853743653221450721012498261, 1.80244748863381436360324815927, 1.87971754577122038881323764731, 2.49998602546216877371964709139, 2.96343353590840836911262861828, 3.09593886255877725277486805576, 3.54663854481281249684859923722, 3.68674720568590558888193442510, 3.93611836399217042815123225186, 3.97798413278449709741089876212, 4.41534783016159892132341730645, 4.52081601147585233230808931139, 5.00840114273274858489610625466, 5.27603629498225512980487398888, 5.33353134253557802817719305476, 5.48123434591653625900793415515, 5.93548737642554884987300680562, 6.13258299604999053346447391076, 6.53818820367562328796983964398, 6.70984283365105840440582208021, 6.90666079725285702805107729196, 7.01064618409550120197734454941, 7.48250272987207871630447717625
