| L(s) = 1 | − 8·9-s + 24·13-s + 24·37-s − 8·49-s − 24·53-s + 30·81-s + 24·89-s − 192·117-s + 28·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 308·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 8/3·9-s + 6.65·13-s + 3.94·37-s − 8/7·49-s − 3.29·53-s + 10/3·81-s + 2.54·89-s − 17.7·117-s + 2.54·121-s + 0.0887·127-s + 0.0873·131-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 + 23.6·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.999595411\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.999595411\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| good | 3 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) | 4.3.a_i_a_bi |
| 7 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) | 4.7.a_i_a_ek |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) | 4.11.a_abc_a_qw |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) | 4.13.ay_ki_acrg_lsw |
| 17 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) | 4.17.a_e_a_wk |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.19.a_acy_a_dfi |
| 23 | $C_2^2$ | \( ( 1 - 28 T^{2} + p^{2} T^{4} )^{2} \) | 4.23.a_ace_a_csw |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.29.a_aem_a_hmc |
| 31 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) | 4.31.a_au_a_czu |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) | 4.37.ay_oa_affs_blso |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.41.a_gi_a_oxy |
| 43 | $C_2^2$ | \( ( 1 + 68 T^{2} + p^{2} T^{4} )^{2} \) | 4.43.a_fg_a_mic |
| 47 | $C_2^2$ | \( ( 1 - 76 T^{2} + p^{2} T^{4} )^{2} \) | 4.47.a_afw_a_pcc |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) | 4.53.y_qm_gya_ciss |
| 59 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) | 4.59.a_u_a_klq |
| 61 | $C_2^2$ | \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \) | 4.61.a_agq_a_vys |
| 67 | $C_2^2$ | \( ( 1 - 28 T^{2} + p^{2} T^{4} )^{2} \) | 4.67.a_ace_a_olm |
| 71 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) | 4.71.a_fk_a_weg |
| 73 | $C_2^2$ | \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{2} \) | 4.73.a_aky_a_btpm |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.79.a_me_a_cdkg |
| 83 | $C_2^2$ | \( ( 1 + 68 T^{2} + p^{2} T^{4} )^{2} \) | 4.83.a_fg_a_bbfu |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) | 4.89.ay_wa_akts_ezco |
| 97 | $C_2^2$ | \( ( 1 - 94 T^{2} + p^{2} T^{4} )^{2} \) | 4.97.a_ahg_a_boxq |
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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
−6.20379663008991781754516158737, −6.01103093383338355090477023290, −5.90418903298411410517204761328, −5.75797345649405027332181330073, −5.41292914992507731070469576595, −5.19068849711690984040190163142, −5.07461139110989358134415801793, −4.58938095386047482430932550205, −4.38816424955758881114743187843, −4.27897820510547305120246446622, −4.02809944012563250579056377440, −3.83763466259965327817303277071, −3.48947873397882113404199557981, −3.30740510619116240005231072683, −3.19235709219894815596362287199, −3.09904450117502718199363740084, −2.99336714960314131840056913127, −2.35692174561620370260324228076, −2.33001068779669216216992423684, −1.67990040537867498909354517828, −1.60487290166128076651333717408, −1.41325869850365159598210890968, −0.924602787024750061447225970878, −0.66313859221663856506581227845, −0.55356135999310187021958065443,
0.55356135999310187021958065443, 0.66313859221663856506581227845, 0.924602787024750061447225970878, 1.41325869850365159598210890968, 1.60487290166128076651333717408, 1.67990040537867498909354517828, 2.33001068779669216216992423684, 2.35692174561620370260324228076, 2.99336714960314131840056913127, 3.09904450117502718199363740084, 3.19235709219894815596362287199, 3.30740510619116240005231072683, 3.48947873397882113404199557981, 3.83763466259965327817303277071, 4.02809944012563250579056377440, 4.27897820510547305120246446622, 4.38816424955758881114743187843, 4.58938095386047482430932550205, 5.07461139110989358134415801793, 5.19068849711690984040190163142, 5.41292914992507731070469576595, 5.75797345649405027332181330073, 5.90418903298411410517204761328, 6.01103093383338355090477023290, 6.20379663008991781754516158737
