| L(s) = 1 | + 4-s + 8·7-s − 12·13-s + 12·19-s − 14·25-s + 8·28-s + 6·31-s + 4·37-s + 16·43-s + 34·49-s − 12·52-s − 64-s − 4·67-s + 24·73-s + 12·76-s + 2·79-s − 96·91-s + 18·97-s − 14·100-s + 4·109-s + 26·121-s + 6·124-s + 127-s + 131-s + 96·133-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 3.02·7-s − 3.32·13-s + 2.75·19-s − 2.79·25-s + 1.51·28-s + 1.07·31-s + 0.657·37-s + 2.43·43-s + 34/7·49-s − 1.66·52-s − 1/8·64-s − 0.488·67-s + 2.80·73-s + 1.37·76-s + 0.225·79-s − 10.0·91-s + 1.82·97-s − 7/5·100-s + 0.383·109-s + 2.36·121-s + 0.538·124-s + 0.0887·127-s + 0.0873·131-s + 8.32·133-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.173388787\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.173388787\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) | |
| 3 | | \( 1 \) | |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) | |
| good | 5 | $C_2^2$ | \( ( 1 + 7 T^{2} + p^{2} T^{4} )^{2} \) | 4.5.a_o_a_dv |
| 11 | $C_2^2$ | \( ( 1 - 13 T^{2} + p^{2} T^{4} )^{2} \) | 4.11.a_aba_a_pv |
| 13 | $C_2^2$ | \( ( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.13.m_di_ro_cvb |
| 17 | $C_2^3$ | \( 1 - 22 T^{2} + 195 T^{4} - 22 p^{2} T^{6} + p^{4} T^{8} \) | 4.17.a_aw_a_hn |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2}( 1 + T + p T^{2} )^{2} \) | 4.19.am_du_axc_enj |
| 23 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) | 4.23.a_au_a_bso |
| 29 | $C_2^3$ | \( 1 + 49 T^{2} + 1560 T^{4} + 49 p^{2} T^{6} + p^{4} T^{8} \) | 4.29.a_bx_a_cia |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) | 4.31.ag_cz_apa_fjw |
| 37 | $C_2^2$ | \( ( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.37.ae_ack_aq_gcp |
| 41 | $C_2^3$ | \( 1 - 34 T^{2} - 525 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \) | 4.41.a_abi_a_auf |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2}( 1 + 5 T + p T^{2} )^{2} \) | 4.43.aq_ec_abnk_ogx |
| 47 | $C_2^3$ | \( 1 - 46 T^{2} - 93 T^{4} - 46 p^{2} T^{6} + p^{4} T^{8} \) | 4.47.a_abu_a_adp |
| 53 | $C_2^3$ | \( 1 + 25 T^{2} - 2184 T^{4} + 25 p^{2} T^{6} + p^{4} T^{8} \) | 4.53.a_z_a_adga |
| 59 | $C_2^3$ | \( 1 - 115 T^{2} + 9744 T^{4} - 115 p^{2} T^{6} + p^{4} T^{8} \) | 4.59.a_ael_a_oku |
| 61 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) | 4.61.a_es_a_qnj |
| 67 | $C_2^2$ | \( ( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.67.e_aes_q_typ |
| 71 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) | 4.71.a_e_a_oxy |
| 73 | $C_2^2$ | \( ( 1 - 12 T + 121 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) | 4.73.ay_ow_agxc_cqnr |
| 79 | $C_2^2$ | \( ( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) | 4.79.ac_afz_ac_bbse |
| 83 | $C_2^3$ | \( 1 - 91 T^{2} + 1392 T^{4} - 91 p^{2} T^{6} + p^{4} T^{8} \) | 4.83.a_adn_a_cbo |
| 89 | $C_2^3$ | \( 1 - 70 T^{2} - 3021 T^{4} - 70 p^{2} T^{6} + p^{4} T^{8} \) | 4.89.a_acs_a_aemf |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 5 T + p T^{2} )^{2} \) | 4.97.as_mr_afxa_cvvo |
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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
−7.15844249501217005562707931393, −6.86400367928751011727549096352, −6.84903794746048614098365437719, −6.26687113198860589411384510313, −6.03451040254870365305277985825, −5.69890620798806598302165388949, −5.67905567976881007511577342079, −5.49290891046844509228026847495, −4.99547342732678598709836581906, −4.95257303754627118076382886331, −4.94885405164377825481164055834, −4.58407989606545191798028235804, −4.28852784004537479981049409992, −4.17712071418802938193831533728, −3.87723588150726914368281782721, −3.39731690440895502415681602182, −3.04005378728758285842249696311, −2.96615293646324386216206869263, −2.31848280304992018754178618450, −2.20846608875597257820842495388, −2.17486142939970264309725197689, −1.84190563836673234661081376830, −1.35454178472985072128998066348, −0.873439987737102373229381500782, −0.59516971859208919818919209654,
0.59516971859208919818919209654, 0.873439987737102373229381500782, 1.35454178472985072128998066348, 1.84190563836673234661081376830, 2.17486142939970264309725197689, 2.20846608875597257820842495388, 2.31848280304992018754178618450, 2.96615293646324386216206869263, 3.04005378728758285842249696311, 3.39731690440895502415681602182, 3.87723588150726914368281782721, 4.17712071418802938193831533728, 4.28852784004537479981049409992, 4.58407989606545191798028235804, 4.94885405164377825481164055834, 4.95257303754627118076382886331, 4.99547342732678598709836581906, 5.49290891046844509228026847495, 5.67905567976881007511577342079, 5.69890620798806598302165388949, 6.03451040254870365305277985825, 6.26687113198860589411384510313, 6.84903794746048614098365437719, 6.86400367928751011727549096352, 7.15844249501217005562707931393
