| L(s) = 1 | − 12·17-s + 20·25-s − 12·41-s − 28·49-s − 4·73-s − 72·89-s − 20·97-s − 72·113-s + 14·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 2.91·17-s + 4·25-s − 1.87·41-s − 4·49-s − 0.468·73-s − 7.63·89-s − 2.03·97-s − 6.77·113-s + 1.27·121-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 + 4·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4970239765\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4970239765\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| good | 5 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.5.a_au_a_fu |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.7.a_bc_a_li |
| 11 | $C_2^3$ | \( 1 - 14 T^{2} + 75 T^{4} - 14 p^{2} T^{6} + p^{4} T^{8} \) | 4.11.a_ao_a_cx |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.13.a_aca_a_bna |
| 17 | $C_2^2$ | \( ( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.17.m_cw_qq_dff |
| 19 | $C_2^3$ | \( 1 + 34 T^{2} + 795 T^{4} + 34 p^{2} T^{6} + p^{4} T^{8} \) | 4.19.a_bi_a_bep |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.23.a_do_a_esc |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.29.a_aem_a_hmc |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.31.a_eu_a_inu |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.37.a_afs_a_mdy |
| 41 | $C_2^2$ | \( ( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.41.m_ba_qq_jjv |
| 43 | $C_2^3$ | \( 1 - 14 T^{2} - 1653 T^{4} - 14 p^{2} T^{6} + p^{4} T^{8} \) | 4.43.a_ao_a_aclp |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.47.a_hg_a_tpu |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.53.a_aie_a_yyg |
| 59 | $C_2^3$ | \( 1 + 82 T^{2} + 3243 T^{4} + 82 p^{2} T^{6} + p^{4} T^{8} \) | 4.59.a_de_a_eut |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.61.a_ajk_a_bhas |
| 67 | $C_2^3$ | \( 1 - 62 T^{2} - 645 T^{4} - 62 p^{2} T^{6} + p^{4} T^{8} \) | 4.67.a_ack_a_ayv |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.71.a_ky_a_bsti |
| 73 | $C_2^2$ | \( ( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.73.e_afe_q_xrn |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.79.a_me_a_cdkg |
| 83 | $C_2^2$ | \( ( 1 + 158 T^{2} + p^{2} T^{4} )^{2} \) | 4.83.a_me_a_cfic |
| 89 | $C_2$ | \( ( 1 + 18 T + p T^{2} )^{4} \) | 4.89.cu_dkm_ckyq_bcjmk |
| 97 | $C_2^2$ | \( ( 1 + 10 T + 3 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) | 4.97.u_ec_cyy_ceoh |
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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
−5.68951684403784384432210902326, −5.58121939675583743691794379928, −5.42554305455222010010252165466, −5.11920402405585130607817942532, −4.96150058220779005019607860610, −4.91582110528020345258185656905, −4.55898799840352488369785733405, −4.36282698737847416310363769215, −4.24630274993356000878944246483, −4.23572679541714320769447782726, −3.99202336420370624824970231567, −3.45014835976629893229285510018, −3.27715146866284772597946065651, −3.17199038828512136087874974095, −2.95954224386891234670455741025, −2.79005849944718192371728581646, −2.43755126054069005645159908713, −2.42879719951160055302093179002, −2.09926111403810818244976995845, −1.58272822554165762556610007985, −1.40455734425236748529902949004, −1.34097892078957790745827216928, −1.18156922366678083546272482981, −0.33326741063559324937090234427, −0.15406605419796194453549698938,
0.15406605419796194453549698938, 0.33326741063559324937090234427, 1.18156922366678083546272482981, 1.34097892078957790745827216928, 1.40455734425236748529902949004, 1.58272822554165762556610007985, 2.09926111403810818244976995845, 2.42879719951160055302093179002, 2.43755126054069005645159908713, 2.79005849944718192371728581646, 2.95954224386891234670455741025, 3.17199038828512136087874974095, 3.27715146866284772597946065651, 3.45014835976629893229285510018, 3.99202336420370624824970231567, 4.23572679541714320769447782726, 4.24630274993356000878944246483, 4.36282698737847416310363769215, 4.55898799840352488369785733405, 4.91582110528020345258185656905, 4.96150058220779005019607860610, 5.11920402405585130607817942532, 5.42554305455222010010252165466, 5.58121939675583743691794379928, 5.68951684403784384432210902326
