| L(s) = 1 | − 2·4-s + 3·16-s + 2·25-s − 16·43-s + 28·49-s − 20·61-s − 4·64-s + 16·79-s − 4·100-s − 16·103-s + 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 32·172-s + 173-s + 179-s + 181-s + 191-s + 193-s − 56·196-s + ⋯ |
| L(s) = 1 | − 4-s + 3/4·16-s + 2/5·25-s − 2.43·43-s + 4·49-s − 2.56·61-s − 1/2·64-s + 1.80·79-s − 2/5·100-s − 1.57·103-s + 4·121-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 + 2.43·172-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s − 4·196-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1844374302\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1844374302\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) | |
| 3 | | \( 1 \) | |
| 13 | | \( 1 \) | |
| good | 5 | $C_2^2$ | \( ( 1 - T^{2} + p^{2} T^{4} )^{2} \) | 4.5.a_ac_a_bz |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.7.a_abc_a_li |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.11.a_abs_a_bby |
| 17 | $C_2^2$ | \( ( 1 + 7 T^{2} + p^{2} T^{4} )^{2} \) | 4.17.a_o_a_yd |
| 19 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) | 4.19.a_u_a_bfq |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.23.a_do_a_esc |
| 29 | $C_2^2$ | \( ( 1 + 31 T^{2} + p^{2} T^{4} )^{2} \) | 4.29.a_ck_a_dxr |
| 31 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) | 4.31.a_abc_a_ddm |
| 37 | $C_2^2$ | \( ( 1 - 71 T^{2} + p^{2} T^{4} )^{2} \) | 4.37.a_afm_a_lnf |
| 41 | $C_2^2$ | \( ( 1 - T^{2} + p^{2} T^{4} )^{2} \) | 4.41.a_ac_a_ezj |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) | 4.43.q_ki_dlg_bdac |
| 47 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) | 4.47.a_dw_a_kgc |
| 53 | $C_2^2$ | \( ( 1 + 79 T^{2} + p^{2} T^{4} )^{2} \) | 4.53.a_gc_a_rod |
| 59 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) | 4.59.a_ca_a_lhu |
| 61 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{4} \) | 4.61.u_pe_gea_cjap |
| 67 | $C_2^2$ | \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \) | 4.67.a_em_a_sgs |
| 71 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) | 4.71.a_e_a_oxy |
| 73 | $C_2^2$ | \( ( 1 - 71 T^{2} + p^{2} T^{4} )^{2} \) | 4.73.a_afm_a_xfv |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) | 4.79.aq_pw_afzs_dafm |
| 83 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) | 4.83.a_abs_a_vco |
| 89 | $C_2^2$ | \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{2} \) | 4.89.a_aky_a_cbgw |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \) | 4.97.a_ae_a_bbvy |
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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
−6.12000014081939564698064372161, −6.06728457454621405807107291642, −5.76581578199832502167762376737, −5.59446827764864869685304643423, −5.19289658345950629398766908218, −5.16236412316579458405159184840, −4.88627419161359656192484390627, −4.83338032848430305386440388928, −4.68641042176570527371273596033, −4.21444146740028128719432277715, −4.04962443455044132317323735561, −3.95803188328121103319379641429, −3.72267697692995649837641009885, −3.47113815400990040063905693320, −3.28171181116782275747627439989, −3.00781956032768884797348530994, −2.59837187741353634336425623454, −2.50516830056014258071061748153, −2.39432705604668066536159847321, −1.75006252060849064487527196888, −1.71183538953490328485373987718, −1.29150886815540818915928524664, −0.994033732854389813728113050317, −0.69592016779145158911274339724, −0.07844166513347921250162925840,
0.07844166513347921250162925840, 0.69592016779145158911274339724, 0.994033732854389813728113050317, 1.29150886815540818915928524664, 1.71183538953490328485373987718, 1.75006252060849064487527196888, 2.39432705604668066536159847321, 2.50516830056014258071061748153, 2.59837187741353634336425623454, 3.00781956032768884797348530994, 3.28171181116782275747627439989, 3.47113815400990040063905693320, 3.72267697692995649837641009885, 3.95803188328121103319379641429, 4.04962443455044132317323735561, 4.21444146740028128719432277715, 4.68641042176570527371273596033, 4.83338032848430305386440388928, 4.88627419161359656192484390627, 5.16236412316579458405159184840, 5.19289658345950629398766908218, 5.59446827764864869685304643423, 5.76581578199832502167762376737, 6.06728457454621405807107291642, 6.12000014081939564698064372161
