| L(s) = 1 | − 4·2-s + 10·4-s − 2·7-s − 20·8-s − 5·9-s + 8·14-s + 35·16-s + 20·18-s − 20·28-s + 12·29-s − 56·32-s − 50·36-s − 2·37-s − 30·47-s − 9·49-s + 40·56-s − 48·58-s − 4·61-s + 10·63-s + 84·64-s − 16·67-s + 100·72-s − 40·73-s + 8·74-s + 4·79-s + 9·81-s + 12·83-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 5·4-s − 0.755·7-s − 7.07·8-s − 5/3·9-s + 2.13·14-s + 35/4·16-s + 4.71·18-s − 3.77·28-s + 2.22·29-s − 9.89·32-s − 8.33·36-s − 0.328·37-s − 4.37·47-s − 9/7·49-s + 5.34·56-s − 6.30·58-s − 0.512·61-s + 1.25·63-s + 21/2·64-s − 1.95·67-s + 11.7·72-s − 4.68·73-s + 0.929·74-s + 0.450·79-s + 81-s + 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{4} \) | |
| 5 | | \( 1 \) | |
| 13 | | \( 1 \) | |
| good | 3 | $C_2^3$ | \( 1 + 5 T^{2} + 16 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} \) | 4.3.a_f_a_q |
| 7 | $D_{4}$ | \( ( 1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) | 4.7.c_n_ba_fs |
| 11 | $D_4$ | \( 1 + 16 T^{2} + 174 T^{4} + 16 p^{2} T^{6} + p^{4} T^{8} \) | 4.11.a_q_a_gs |
| 17 | $D_4\times C_2$ | \( 1 + 25 T^{2} + 528 T^{4} + 25 p^{2} T^{6} + p^{4} T^{8} \) | 4.17.a_z_a_ui |
| 19 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) | 4.19.a_ca_a_cbu |
| 23 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) | 4.23.a_e_a_bow |
| 29 | $D_{4}$ | \( ( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.29.am_ea_abdc_hhm |
| 31 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) | 4.31.a_dw_a_goc |
| 37 | $D_{4}$ | \( ( 1 + T + p T^{3} + p^{2} T^{4} )^{2} \) | 4.37.c_b_cw_eee |
| 41 | $D_4\times C_2$ | \( 1 + 52 T^{2} + 1926 T^{4} + 52 p^{2} T^{6} + p^{4} T^{8} \) | 4.41.a_ca_a_cwc |
| 43 | $D_4\times C_2$ | \( 1 + 85 T^{2} + 3648 T^{4} + 85 p^{2} T^{6} + p^{4} T^{8} \) | 4.43.a_dh_a_fki |
| 47 | $D_{4}$ | \( ( 1 + 15 T + 142 T^{2} + 15 p T^{3} + p^{2} T^{4} )^{2} \) | 4.47.be_tp_ikc_cpqy |
| 53 | $D_4\times C_2$ | \( 1 + 184 T^{2} + 13950 T^{4} + 184 p^{2} T^{6} + p^{4} T^{8} \) | 4.53.a_hc_a_uqo |
| 59 | $C_2^2$ | \( ( 1 + 74 T^{2} + p^{2} T^{4} )^{2} \) | 4.59.a_fs_a_skk |
| 61 | $D_{4}$ | \( ( 1 + 2 T + 90 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.61.e_hc_xg_xso |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) | 4.67.q_oa_fdo_chgo |
| 71 | $D_4\times C_2$ | \( 1 + 109 T^{2} + 7896 T^{4} + 109 p^{2} T^{6} + p^{4} T^{8} \) | 4.71.a_ef_a_lrs |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) | 4.73.bo_bii_swu_hjrq |
| 79 | $D_{4}$ | \( ( 1 - 2 T + 126 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.79.ae_jw_abfo_bqxa |
| 83 | $D_{4}$ | \( ( 1 - 6 T + 142 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.83.am_mi_adzw_chbi |
| 89 | $D_4\times C_2$ | \( 1 + 280 T^{2} + 34254 T^{4} + 280 p^{2} T^{6} + p^{4} T^{8} \) | 4.89.a_ku_a_byrm |
| 97 | $D_{4}$ | \( ( 1 + 2 T + 162 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.97.e_mq_bnw_cpva |
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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
−6.02486752715796225740238449107, −5.79711519560780346492074366205, −5.51978875209916430181453759208, −5.43770709401926026291630519226, −5.18484718498144683031395584496, −4.87587263847624816992014972598, −4.80166769099712196827275717587, −4.62247829759366711522783419826, −4.49687556462840375380241076876, −4.01576462632616592774216914823, −3.86946294201935690471533228784, −3.60666513487409821007672997369, −3.33582818191721729292146432678, −3.02617188205881403046050796644, −3.01526554152335657197715478194, −2.94549764019415917482934897336, −2.94166830332141332145840846094, −2.47853571552205990360268336238, −2.18752170937705188713179368957, −2.01654657980951131757004724244, −1.67867441121874570178700512616, −1.66563605111273571618298122307, −1.15858172498299653770357819075, −1.02542880852711061082041898181, −0.944206183046667200031165653308, 0, 0, 0, 0,
0.944206183046667200031165653308, 1.02542880852711061082041898181, 1.15858172498299653770357819075, 1.66563605111273571618298122307, 1.67867441121874570178700512616, 2.01654657980951131757004724244, 2.18752170937705188713179368957, 2.47853571552205990360268336238, 2.94166830332141332145840846094, 2.94549764019415917482934897336, 3.01526554152335657197715478194, 3.02617188205881403046050796644, 3.33582818191721729292146432678, 3.60666513487409821007672997369, 3.86946294201935690471533228784, 4.01576462632616592774216914823, 4.49687556462840375380241076876, 4.62247829759366711522783419826, 4.80166769099712196827275717587, 4.87587263847624816992014972598, 5.18484718498144683031395584496, 5.43770709401926026291630519226, 5.51978875209916430181453759208, 5.79711519560780346492074366205, 6.02486752715796225740238449107
