| L(s) = 1 | − 4·3-s − 4·4-s + 4·7-s + 8·9-s − 4·11-s + 16·12-s + 4·13-s + 12·16-s − 16·21-s − 12·23-s + 8·25-s − 16·27-s − 16·28-s + 12·29-s + 16·33-s − 32·36-s + 16·37-s − 16·39-s + 24·41-s + 4·43-s + 16·44-s + 12·47-s − 48·48-s + 8·49-s − 16·52-s − 24·53-s + 32·63-s + ⋯ |
| L(s) = 1 | − 2.30·3-s − 2·4-s + 1.51·7-s + 8/3·9-s − 1.20·11-s + 4.61·12-s + 1.10·13-s + 3·16-s − 3.49·21-s − 2.50·23-s + 8/5·25-s − 3.07·27-s − 3.02·28-s + 2.22·29-s + 2.78·33-s − 5.33·36-s + 2.63·37-s − 2.56·39-s + 3.74·41-s + 0.609·43-s + 2.41·44-s + 1.75·47-s − 6.92·48-s + 8/7·49-s − 2.21·52-s − 3.29·53-s + 4.03·63-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
| \(L(1)\) |
\(\approx\) |
\(0.6099961035\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6099961035\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) | |
| 11 | $C_1$ | \( ( 1 + T )^{4} \) | |
| good | 3 | $D_4\times C_2$ | \( 1 + 4 T + 8 T^{2} + 16 T^{3} + 31 T^{4} + 16 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.3.e_i_q_bf |
| 7 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} - p^{2} T^{4} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.7.ae_i_a_abx |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) | 4.13.ae_i_aci_re |
| 17 | $C_2^3$ | \( 1 + 511 T^{4} + p^{4} T^{8} \) | 4.17.a_a_a_tr |
| 19 | $D_4\times C_2$ | \( 1 - 22 T^{2} + 195 T^{4} - 22 p^{2} T^{6} + p^{4} T^{8} \) | 4.19.a_aw_a_hn |
| 23 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} + 300 T^{3} + 1246 T^{4} + 300 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.23.m_cu_lo_bvy |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{4} \) | 4.29.am_go_absi_mfr |
| 31 | $D_4\times C_2$ | \( 1 - 70 T^{2} + 2499 T^{4} - 70 p^{2} T^{6} + p^{4} T^{8} \) | 4.31.a_acs_a_dsd |
| 37 | $D_4\times C_2$ | \( 1 - 16 T + 128 T^{2} - 960 T^{3} + 6671 T^{4} - 960 p T^{5} + 128 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) | 4.37.aq_ey_abky_jwp |
| 41 | $D_{4}$ | \( ( 1 - 12 T + 100 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) | 4.41.ay_ng_afae_blgc |
| 43 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} - 36 T^{3} - 994 T^{4} - 36 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.43.ae_i_abk_abmg |
| 47 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 12 T^{3} - 2114 T^{4} - 12 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.47.am_cu_am_addi |
| 53 | $D_4\times C_2$ | \( 1 + 24 T + 288 T^{2} + 2976 T^{3} + 25711 T^{4} + 2976 p T^{5} + 288 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) | 4.53.y_lc_ekm_bmax |
| 59 | $C_2^2$ | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \) | 4.59.a_abo_a_kxe |
| 61 | $C_2^2$ | \( ( 1 - 41 T^{2} + p^{2} T^{4} )^{2} \) | 4.61.a_ade_a_nmx |
| 67 | $C_2^2$ | \( ( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.67.i_bg_xc_qog |
| 71 | $D_4\times C_2$ | \( 1 - 262 T^{2} + 27171 T^{4} - 262 p^{2} T^{6} + p^{4} T^{8} \) | 4.71.a_akc_a_bofb |
| 73 | $D_4\times C_2$ | \( 1 - 16 T + 128 T^{2} - 1104 T^{3} + 9506 T^{4} - 1104 p T^{5} + 128 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) | 4.73.aq_ey_abqm_obq |
| 79 | $D_{4}$ | \( ( 1 + 8 T + 156 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) | 4.79.q_om_foq_crla |
| 83 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 12 T^{3} - 6722 T^{4} - 12 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.83.am_cu_am_ajyo |
| 89 | $D_4\times C_2$ | \( 1 - 322 T^{2} + 41475 T^{4} - 322 p^{2} T^{6} + p^{4} T^{8} \) | 4.89.a_amk_a_cjjf |
| 97 | $D_4\times C_2$ | \( 1 + 32 T + 512 T^{2} + 6048 T^{3} + 62978 T^{4} + 6048 p T^{5} + 512 p^{2} T^{6} + 32 p^{3} T^{7} + p^{4} T^{8} \) | 4.97.bg_ts_iyq_dpeg |
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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
−8.051076711533365674798104694702, −7.84171272433279562384436236478, −7.71126005466380576697455608582, −7.58819102154261952916683637438, −7.06272880802252015943560819018, −6.50848703912025781519892482402, −6.42027055650418345945871013124, −6.15386849945111290714895216001, −6.05931266043645001278161133138, −5.55942415423950001151609940576, −5.44612379325015000149249556496, −5.35079006932615608698285845195, −5.21900736292781123565082478594, −4.59216502637656171345415021234, −4.38506301866316082008065667316, −4.22110258526589909350065469151, −4.15559105123702208268564746913, −4.08670941666120269166857360343, −3.13111195441901863767838748539, −2.79785505050498424481536048233, −2.63203553634285890048727447310, −1.79099679261792904264095088516, −1.32202211940042443126591921155, −0.844515752554539964653167721917, −0.52873515043825231688631598629,
0.52873515043825231688631598629, 0.844515752554539964653167721917, 1.32202211940042443126591921155, 1.79099679261792904264095088516, 2.63203553634285890048727447310, 2.79785505050498424481536048233, 3.13111195441901863767838748539, 4.08670941666120269166857360343, 4.15559105123702208268564746913, 4.22110258526589909350065469151, 4.38506301866316082008065667316, 4.59216502637656171345415021234, 5.21900736292781123565082478594, 5.35079006932615608698285845195, 5.44612379325015000149249556496, 5.55942415423950001151609940576, 6.05931266043645001278161133138, 6.15386849945111290714895216001, 6.42027055650418345945871013124, 6.50848703912025781519892482402, 7.06272880802252015943560819018, 7.58819102154261952916683637438, 7.71126005466380576697455608582, 7.84171272433279562384436236478, 8.051076711533365674798104694702
