| L(s) = 1 | + 4·5-s + 6·9-s − 10·25-s − 12·37-s + 24·45-s − 12·49-s + 8·53-s + 9·81-s − 4·89-s + 28·97-s − 20·113-s + 10·121-s − 80·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4·169-s + 173-s + 179-s + 181-s − 48·185-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 2·9-s − 2·25-s − 1.97·37-s + 3.57·45-s − 1.71·49-s + 1.09·53-s + 81-s − 0.423·89-s + 2.84·97-s − 1.88·113-s + 0.909·121-s − 7.15·125-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 + 4/13·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s − 3.52·185-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.706851310\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.706851310\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) | |
| good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )^{2}( 1 + p T + p T^{2} )^{2} \) | 4.3.a_ag_a_bb |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) | 4.5.ae_ba_acm_id |
| 7 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) | 4.7.a_m_a_fe |
| 13 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) | 4.13.a_ae_a_ne |
| 17 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) | 4.17.a_au_a_bac |
| 19 | $C_2^2$ | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \) | 4.19.a_ci_a_ckk |
| 23 | $C_2^2$ | \( ( 1 - 19 T^{2} + p^{2} T^{4} )^{2} \) | 4.23.a_abm_a_ccp |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.29.a_aem_a_hmc |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2}( 1 + 11 T + p T^{2} )^{2} \) | 4.31.a_aeo_a_hzv |
| 37 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) | 4.37.m_hu_cdk_set |
| 41 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) | 4.41.a_aem_a_jys |
| 43 | $C_2^2$ | \( ( 1 + 54 T^{2} + p^{2} T^{4} )^{2} \) | 4.43.a_ee_a_juk |
| 47 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) | 4.47.a_agi_a_qmo |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) | 4.53.ai_jc_abye_bcss |
| 59 | $C_2^2$ | \( ( 1 - 115 T^{2} + p^{2} T^{4} )^{2} \) | 4.59.a_aiw_a_bdwl |
| 61 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) | 4.61.a_aca_a_mag |
| 67 | $C_2^2$ | \( ( 1 - 59 T^{2} + p^{2} T^{4} )^{2} \) | 4.67.a_aeo_a_slf |
| 71 | $C_2^2$ | \( ( 1 + 5 T^{2} + p^{2} T^{4} )^{2} \) | 4.71.a_k_a_oyt |
| 73 | $C_2^2$ | \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{2} \) | 4.73.a_ajk_a_bluk |
| 79 | $C_2^2$ | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \) | 4.79.a_ci_a_tus |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.83.a_mu_a_cjdu |
| 89 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) | 4.89.e_ny_bpg_ctxb |
| 97 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{4} \) | 4.97.abc_bag_aoce_gplj |
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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
−7.43002867319010855757193659007, −7.20032664574353771875565430672, −7.11949067568228553515784966260, −6.79415779039305815821390541562, −6.58208339815802889882460182061, −6.21947640678575001360658103209, −6.20694997345248234279900260599, −5.82434611234115227194851696855, −5.65308460365124071288827623972, −5.44269579664047566980673543436, −5.16024127731067142380369532456, −4.86234610097455460762828675567, −4.69414664880471483210832050064, −4.18048214163117668166960166675, −4.06772710156915933602609611853, −3.95547387967267418484496096093, −3.50961743496104643758950889906, −3.08495771562834710820539576194, −3.04002534353701236805099933532, −2.21472896766727563272160464990, −2.04235409454328899982711945799, −1.98300052056500073735285023251, −1.54207988866898820640457981296, −1.34787714835642310671452480849, −0.48308235384035131426042242840,
0.48308235384035131426042242840, 1.34787714835642310671452480849, 1.54207988866898820640457981296, 1.98300052056500073735285023251, 2.04235409454328899982711945799, 2.21472896766727563272160464990, 3.04002534353701236805099933532, 3.08495771562834710820539576194, 3.50961743496104643758950889906, 3.95547387967267418484496096093, 4.06772710156915933602609611853, 4.18048214163117668166960166675, 4.69414664880471483210832050064, 4.86234610097455460762828675567, 5.16024127731067142380369532456, 5.44269579664047566980673543436, 5.65308460365124071288827623972, 5.82434611234115227194851696855, 6.20694997345248234279900260599, 6.21947640678575001360658103209, 6.58208339815802889882460182061, 6.79415779039305815821390541562, 7.11949067568228553515784966260, 7.20032664574353771875565430672, 7.43002867319010855757193659007
