| L(s) = 1 | + 10·9-s + 2·25-s − 28·49-s + 57·81-s + 36·89-s − 68·97-s − 84·113-s + 22·121-s + ⋯ |
| L(s) = 1 | + 10/3·9-s + 2/5·25-s − 4·49-s + 19/3·81-s + 3.81·89-s − 6.90·97-s − 7.90·113-s + 2·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7555589859\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7555589859\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) | |
| good | 3 | $C_2^2$ | \( ( 1 - 5 T^{2} + p^{2} T^{4} )^{2} \) | 4.3.a_ak_a_br |
| 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_bc_a_li |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.13.a_ca_a_bna |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.17.a_acq_a_cos |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.19.a_acy_a_dfi |
| 23 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2}( 1 + 9 T + p T^{2} )^{2} \) | 4.23.a_acs_a_djv |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.29.a_em_a_hmc |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2}( 1 + 5 T + p T^{2} )^{2} \) | 4.31.a_cw_a_ewp |
| 37 | $C_2^2$ | \( ( 1 - 25 T^{2} + p^{2} T^{4} )^{2} \) | 4.37.a_aby_a_ezj |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.41.a_agi_a_oxy |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.43.a_agq_a_qks |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) | 4.47.a_adw_a_kgc |
| 53 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) | 4.53.a_afk_a_poo |
| 59 | $C_2^2$ | \( ( 1 + 107 T^{2} + p^{2} T^{4} )^{2} \) | 4.59.a_ig_a_bbgd |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.61.a_jk_a_bhas |
| 67 | $C_2^2$ | \( ( 1 + 35 T^{2} + p^{2} T^{4} )^{2} \) | 4.67.a_cs_a_pcl |
| 71 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2}( 1 + 3 T + p T^{2} )^{2} \) | 4.71.a_kg_a_bpcd |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.73.a_alg_a_bvhu |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.79.a_me_a_cdkg |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.83.a_amu_a_cjdu |
| 89 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{4} \) | 4.89.abk_bgk_asnw_hzzn |
| 97 | $C_2$ | \( ( 1 + 17 T + p T^{2} )^{4} \) | 4.97.cq_ddq_cgiy_bbcrz |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.28112950243654580729310155764, −6.27495287544410493997397845685, −5.85770081209442332159804974212, −5.41939771897886739534084915664, −5.40332961934681688588666639330, −5.19291021906610373757256407671, −4.93836560042813342069750601640, −4.72575608159544033466697369664, −4.56400322265379211794966593772, −4.51761137104174719945196121106, −3.94794735948926606505467989861, −3.86754329358391859924914604225, −3.84820595234986949428889067284, −3.79919458488347069675570406478, −3.09277439014469618477807125665, −3.08465445247433503966741023318, −2.76887423399141173296664278410, −2.52183389423270904862779536715, −2.12515996259660670686997468995, −1.86137532145363574658900962186, −1.52562431655728173306011074383, −1.41724734698628641674531263699, −1.21986679994038940261050732527, −0.904865739246197683215103865476, −0.11684236313958310069999527090,
0.11684236313958310069999527090, 0.904865739246197683215103865476, 1.21986679994038940261050732527, 1.41724734698628641674531263699, 1.52562431655728173306011074383, 1.86137532145363574658900962186, 2.12515996259660670686997468995, 2.52183389423270904862779536715, 2.76887423399141173296664278410, 3.08465445247433503966741023318, 3.09277439014469618477807125665, 3.79919458488347069675570406478, 3.84820595234986949428889067284, 3.86754329358391859924914604225, 3.94794735948926606505467989861, 4.51761137104174719945196121106, 4.56400322265379211794966593772, 4.72575608159544033466697369664, 4.93836560042813342069750601640, 5.19291021906610373757256407671, 5.40332961934681688588666639330, 5.41939771897886739534084915664, 5.85770081209442332159804974212, 6.27495287544410493997397845685, 6.28112950243654580729310155764
