| L(s) = 1 | + 2·4-s − 20·19-s + 4·25-s + 4·31-s + 20·37-s − 8·64-s − 40·76-s + 8·100-s + 4·103-s + 20·109-s − 20·121-s + 8·124-s + 127-s + 131-s + 137-s + 139-s + 40·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | + 4-s − 4.58·19-s + 4/5·25-s + 0.718·31-s + 3.28·37-s − 64-s − 4.58·76-s + 4/5·100-s + 0.394·103-s + 1.91·109-s − 1.81·121-s + 0.718·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 3.28·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.153·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.449378192\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.449378192\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) | |
| 3 | | \( 1 \) | |
| 7 | | \( 1 \) | |
| good | 5 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) | 4.5.a_ae_a_cc |
| 11 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) | 4.11.a_u_a_ne |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2}( 1 + 5 T + p T^{2} )^{2} \) | 4.13.a_c_a_nb |
| 17 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) | 4.17.a_aca_a_bwg |
| 19 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{4} \) | 4.19.u_is_clc_mop |
| 23 | $C_2^2$ | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \) | 4.23.a_acy_a_dsg |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.29.a_em_a_hmc |
| 31 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) | 4.31.ae_fa_aom_jcd |
| 37 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{4} \) | 4.37.au_lm_aeaq_bdmx |
| 41 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) | 4.41.a_adw_a_irm |
| 43 | $C_2^2$ | \( ( 1 - 59 T^{2} + p^{2} T^{4} )^{2} \) | 4.43.a_aeo_a_kqd |
| 47 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) | 4.47.a_fk_a_nuk |
| 53 | $C_2^2$ | \( ( 1 + 82 T^{2} + p^{2} T^{4} )^{2} \) | 4.53.a_gi_a_sgs |
| 59 | $C_2^2$ | \( ( 1 + 94 T^{2} + p^{2} T^{4} )^{2} \) | 4.59.a_hg_a_xjq |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \) | 4.61.a_afs_a_tcw |
| 67 | $C_2^2$ | \( ( 1 - 59 T^{2} + p^{2} T^{4} )^{2} \) | 4.67.a_aeo_a_slf |
| 71 | $C_2^2$ | \( ( 1 - 134 T^{2} + p^{2} T^{4} )^{2} \) | 4.71.a_aki_a_bpmk |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )^{2}( 1 + 17 T + p T^{2} )^{2} \) | 4.73.a_ala_a_bual |
| 79 | $C_2^2$ | \( ( 1 - 155 T^{2} + p^{2} T^{4} )^{2} \) | 4.79.a_aly_a_ccad |
| 83 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) | 4.83.a_adw_a_ycc |
| 89 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) | 4.89.a_adw_a_bbdm |
| 97 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.97.a_aoy_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
−6.47169478361133191405892828482, −6.45910515435638773240697819269, −6.25326278828930742358600079353, −6.20852886570848871896425790395, −5.73914660609603875801311357219, −5.66011602225833131936998590975, −5.61850897855876204226254071666, −4.86224414335583847156524388372, −4.68537219181453053493284655361, −4.67111419823387410073788205638, −4.64066978945367483892715451175, −4.09986988919614786639420109537, −4.00062066839684043680095118053, −3.90848446089770691695777244188, −3.53216540539354437815959230264, −2.99346417852423039442607799818, −2.84500704331617011819116857711, −2.71591873381307932437364859599, −2.40848024512063413696628987116, −2.15517206375599919105648209805, −1.85936860019013132032101720244, −1.77458789953993893363670907177, −1.20762220676133789044944100252, −0.67024818240237012750038586797, −0.39748347633630039494090043999,
0.39748347633630039494090043999, 0.67024818240237012750038586797, 1.20762220676133789044944100252, 1.77458789953993893363670907177, 1.85936860019013132032101720244, 2.15517206375599919105648209805, 2.40848024512063413696628987116, 2.71591873381307932437364859599, 2.84500704331617011819116857711, 2.99346417852423039442607799818, 3.53216540539354437815959230264, 3.90848446089770691695777244188, 4.00062066839684043680095118053, 4.09986988919614786639420109537, 4.64066978945367483892715451175, 4.67111419823387410073788205638, 4.68537219181453053493284655361, 4.86224414335583847156524388372, 5.61850897855876204226254071666, 5.66011602225833131936998590975, 5.73914660609603875801311357219, 6.20852886570848871896425790395, 6.25326278828930742358600079353, 6.45910515435638773240697819269, 6.47169478361133191405892828482
