| L(s) = 1 | + 3·7-s + 10·11-s + 16·23-s + 9·25-s + 2·43-s + 7·49-s + 30·77-s + 47·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 48·161-s + 163-s + 167-s − 52·169-s + 173-s + 27·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | + 1.13·7-s + 3.01·11-s + 3.33·23-s + 9/5·25-s + 0.304·43-s + 49-s + 3.41·77-s + 4.27·121-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 + 3.78·161-s + 0.0783·163-s + 0.0773·167-s − 4·169-s + 0.0760·173-s + 2.04·175-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(16.48807442\) |
| \(L(\frac12)\) |
\(\approx\) |
\(16.48807442\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 7 | $C_2^2$ | \( 1 - 3 T + 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) | |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) | |
| good | 5 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} )( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} ) \) | 4.5.a_aj_a_ce |
| 11 | $C_2^2$ | \( ( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) | 4.11.ak_cb_ajq_bma |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.13.a_ca_a_bna |
| 17 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 7 T + 32 T^{2} - 7 p T^{3} + p^{2} T^{4} )( 1 + 7 T + 32 T^{2} + 7 p T^{3} + p^{2} T^{4} ) \) | 4.17.a_p_a_acm |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) | 4.23.aq_hg_acai_lpu |
| 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 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.41.a_gi_a_oxy |
| 43 | $C_2^2$ | \( ( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) | 4.43.ac_adf_ac_ifk |
| 47 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 13 T + 122 T^{2} - 13 p T^{3} + p^{2} T^{4} )( 1 + 13 T + 122 T^{2} + 13 p T^{3} + p^{2} T^{4} ) \) | 4.47.a_cx_a_fbk |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.53.a_aie_a_yyg |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.59.a_jc_a_bexi |
| 61 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 15 T + 164 T^{2} - 15 p T^{3} + p^{2} T^{4} )( 1 + 15 T + 164 T^{2} + 15 p T^{3} + p^{2} T^{4} ) \) | 4.61.a_dz_a_key |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.67.a_aki_a_bnvy |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.71.a_aky_a_bsti |
| 73 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 11 T + 48 T^{2} - 11 p T^{3} + p^{2} T^{4} )( 1 + 11 T + 48 T^{2} + 11 p T^{3} + p^{2} T^{4} ) \) | 4.73.a_az_a_agyy |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.79.a_ame_a_cdkg |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2}( 1 + 16 T + p T^{2} )^{2} \) | 4.83.a_agy_a_bgjm |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.89.a_ns_a_cshy |
| 97 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.97.a_oy_a_dfni |
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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
−5.65417724935913037807038406322, −5.56289522326117271028742199441, −5.46625849790942345736447618095, −5.21197785142666620256651559969, −5.16688347224187788139589573806, −4.71517969853819842153244223138, −4.65778883781367992477666545471, −4.55663572264720897243790232812, −4.32027918537686298905488285093, −3.99815011833789226805863794297, −3.97260164112803801302624274965, −3.69851669577284120057481559023, −3.40768025985537605821831967829, −3.20154230418691952612555133637, −2.97304968413595728529221890848, −2.92075952235850943800334646045, −2.46356398998730988096788220560, −2.39060188148230673994004127022, −1.94759443545029728960742328540, −1.56793139248720687484785125544, −1.52401566350234417316584045461, −1.17624772787336207381416259242, −1.13214971105214111021625516753, −0.62313692861065235664004605358, −0.61368694693971317305710204511,
0.61368694693971317305710204511, 0.62313692861065235664004605358, 1.13214971105214111021625516753, 1.17624772787336207381416259242, 1.52401566350234417316584045461, 1.56793139248720687484785125544, 1.94759443545029728960742328540, 2.39060188148230673994004127022, 2.46356398998730988096788220560, 2.92075952235850943800334646045, 2.97304968413595728529221890848, 3.20154230418691952612555133637, 3.40768025985537605821831967829, 3.69851669577284120057481559023, 3.97260164112803801302624274965, 3.99815011833789226805863794297, 4.32027918537686298905488285093, 4.55663572264720897243790232812, 4.65778883781367992477666545471, 4.71517969853819842153244223138, 5.16688347224187788139589573806, 5.21197785142666620256651559969, 5.46625849790942345736447618095, 5.56289522326117271028742199441, 5.65417724935913037807038406322
