| L(s) = 1 | − 8·25-s − 32·37-s + 14·49-s + 40·109-s + 40·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
| L(s) = 1 | − 8/5·25-s − 5.26·37-s + 2·49-s + 3.83·109-s + 3.63·121-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 + 0.0783·163-s + 0.0773·167-s + 4·169-s + 0.0760·173-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 + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.174998881\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.174998881\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) | |
| good | 5 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) | 4.5.a_i_a_co |
| 11 | $C_2^2$ | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \) | 4.11.a_abo_a_ys |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.13.a_aca_a_bna |
| 17 | $C_2^2$ | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \) | 4.17.a_abo_a_blq |
| 19 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) | 4.19.a_u_a_bfq |
| 23 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) | 4.23.a_i_a_bpi |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.29.a_em_a_hmc |
| 31 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) | 4.31.a_adw_a_goc |
| 37 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) | 4.37.bg_um_ihk_cigk |
| 41 | $C_2^2$ | \( ( 1 - 68 T^{2} + p^{2} T^{4} )^{2} \) | 4.41.a_afg_a_lve |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.43.a_agq_a_qks |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.47.a_hg_a_tpu |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.53.a_ie_a_yyg |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.59.a_jc_a_bexi |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.61.a_ajk_a_bhas |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.67.a_aki_a_bnvy |
| 71 | $C_2^2$ | \( ( 1 + 100 T^{2} + p^{2} T^{4} )^{2} \) | 4.71.a_hs_a_bdsk |
| 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_ame_a_cdkg |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.83.a_mu_a_cjdu |
| 89 | $C_2^2$ | \( ( 1 + 172 T^{2} + p^{2} T^{4} )^{2} \) | 4.89.a_ng_a_cpfe |
| 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
−5.81088808704040946129232956336, −5.70377137805178776040046582946, −5.69362341881838239839573052327, −5.44707558162084753542220249391, −5.20264710316573080057525216717, −4.97706734849068305613268718558, −4.72770512284783542393552312106, −4.63301888400959383065638028255, −4.42571674796817603135176337545, −4.10930610993183011012180652743, −3.90391262661311765428432367149, −3.68026548589517879759074379429, −3.46120051996848952696778801661, −3.40969383147543061252171777327, −3.08360337822523577240644749794, −3.04047936390452240037395711548, −2.39662364549712851737278020790, −2.34283592742943111431231337067, −2.11782340742750314979975905541, −1.72504524606073368687065853470, −1.72004490357957861501072135939, −1.46790695838806171335219182728, −0.867090019759531006383878658619, −0.56330182244325714126151402218, −0.26469169943726369720336259473,
0.26469169943726369720336259473, 0.56330182244325714126151402218, 0.867090019759531006383878658619, 1.46790695838806171335219182728, 1.72004490357957861501072135939, 1.72504524606073368687065853470, 2.11782340742750314979975905541, 2.34283592742943111431231337067, 2.39662364549712851737278020790, 3.04047936390452240037395711548, 3.08360337822523577240644749794, 3.40969383147543061252171777327, 3.46120051996848952696778801661, 3.68026548589517879759074379429, 3.90391262661311765428432367149, 4.10930610993183011012180652743, 4.42571674796817603135176337545, 4.63301888400959383065638028255, 4.72770512284783542393552312106, 4.97706734849068305613268718558, 5.20264710316573080057525216717, 5.44707558162084753542220249391, 5.69362341881838239839573052327, 5.70377137805178776040046582946, 5.81088808704040946129232956336
