| L(s) = 1 | − 4·7-s − 4·13-s + 4·19-s − 14·25-s + 4·31-s − 12·43-s + 10·49-s − 32·61-s − 8·67-s − 8·73-s − 24·79-s + 16·91-s − 32·97-s − 16·103-s − 28·121-s + 127-s + 131-s − 16·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 32·169-s + 173-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 1.10·13-s + 0.917·19-s − 2.79·25-s + 0.718·31-s − 1.82·43-s + 10/7·49-s − 4.09·61-s − 0.977·67-s − 0.936·73-s − 2.70·79-s + 1.67·91-s − 3.24·97-s − 1.57·103-s − 2.54·121-s + 0.0887·127-s + 0.0873·131-s − 1.38·133-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 − 2.46·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 3 | | \( 1 \) | |
| 7 | $C_1$ | \( ( 1 + T )^{4} \) | |
| 19 | $C_1$ | \( ( 1 - T )^{4} \) | |
| good | 5 | $D_4\times C_2$ | \( 1 + 14 T^{2} + 94 T^{4} + 14 p^{2} T^{6} + p^{4} T^{8} \) | 4.5.a_o_a_dq |
| 11 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) | 4.11.a_bc_a_qw |
| 13 | $D_{4}$ | \( ( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.13.e_bw_fk_bjq |
| 17 | $D_4\times C_2$ | \( 1 + 54 T^{2} + 1262 T^{4} + 54 p^{2} T^{6} + p^{4} T^{8} \) | 4.17.a_cc_a_bwo |
| 23 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) | 4.23.a_cy_a_dsg |
| 29 | $D_4\times C_2$ | \( 1 - 10 T^{2} + 1582 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} \) | 4.29.a_ak_a_ciw |
| 31 | $D_{4}$ | \( ( 1 - 2 T + 18 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.31.ae_bo_aho_dry |
| 37 | $C_2^2$ | \( ( 1 + 54 T^{2} + p^{2} T^{4} )^{2} \) | 4.37.a_ee_a_ijm |
| 41 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) | 4.41.a_u_a_fde |
| 43 | $D_{4}$ | \( ( 1 + 6 T + 90 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.43.m_ii_cjk_waw |
| 47 | $D_4\times C_2$ | \( 1 + 142 T^{2} + 9414 T^{4} + 142 p^{2} T^{6} + p^{4} T^{8} \) | 4.47.a_fm_a_nyc |
| 53 | $D_4\times C_2$ | \( 1 + 62 T^{2} + 3454 T^{4} + 62 p^{2} T^{6} + p^{4} T^{8} \) | 4.53.a_ck_a_fcw |
| 59 | $D_4\times C_2$ | \( 1 + 12 T^{2} - 4522 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} \) | 4.59.a_m_a_agry |
| 61 | $C_4$ | \( ( 1 + 16 T + 166 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) | 4.61.bg_wq_ktk_dtzi |
| 67 | $D_{4}$ | \( ( 1 + 4 T - 42 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.67.i_acq_hs_tbq |
| 71 | $D_4\times C_2$ | \( 1 + 118 T^{2} + 8118 T^{4} + 118 p^{2} T^{6} + p^{4} T^{8} \) | 4.71.a_eo_a_mag |
| 73 | $D_{4}$ | \( ( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.73.i_ga_bsa_bamg |
| 79 | $D_{4}$ | \( ( 1 + 12 T + 174 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) | 4.79.y_sy_izo_dsxq |
| 83 | $D_4\times C_2$ | \( 1 + 246 T^{2} + 28502 T^{4} + 246 p^{2} T^{6} + p^{4} T^{8} \) | 4.83.a_jm_a_bqeg |
| 89 | $C_2^2$ | \( ( 1 + 106 T^{2} + p^{2} T^{4} )^{2} \) | 4.89.a_ie_a_bobm |
| 97 | $D_{4}$ | \( ( 1 + 16 T + 238 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) | 4.97.bg_bce_pwi_hdco |
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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
−6.30924969696008156841077651875, −5.94926427854162062592795687765, −5.92525980389471972325895035321, −5.59888776590281603955370860073, −5.55540532522430471845560115726, −5.20394621947103789740234746554, −5.15763608087347194700895880092, −5.00133336429595421938721905192, −4.52175428633810275897515521388, −4.43351066840150998649827582268, −4.25152600819746338855200793057, −4.03233584795829395532867708854, −4.01259032119415004015264693154, −3.49214986591168343188766404571, −3.35196890872376221500329790374, −3.29834283099864060341770962250, −3.08182528017810445194770658421, −2.60799913722677864600904846178, −2.54446694756704912200377162084, −2.47338784824763596029298995886, −2.27152978849208383081721199037, −1.51033746457637134319415699876, −1.50109432184343539127572684278, −1.29759661102278597750703260628, −1.21917810181995869676589755067, 0, 0, 0, 0,
1.21917810181995869676589755067, 1.29759661102278597750703260628, 1.50109432184343539127572684278, 1.51033746457637134319415699876, 2.27152978849208383081721199037, 2.47338784824763596029298995886, 2.54446694756704912200377162084, 2.60799913722677864600904846178, 3.08182528017810445194770658421, 3.29834283099864060341770962250, 3.35196890872376221500329790374, 3.49214986591168343188766404571, 4.01259032119415004015264693154, 4.03233584795829395532867708854, 4.25152600819746338855200793057, 4.43351066840150998649827582268, 4.52175428633810275897515521388, 5.00133336429595421938721905192, 5.15763608087347194700895880092, 5.20394621947103789740234746554, 5.55540532522430471845560115726, 5.59888776590281603955370860073, 5.92525980389471972325895035321, 5.94926427854162062592795687765, 6.30924969696008156841077651875
