| L(s) = 1 | + 4·16-s + 16·19-s + 22·31-s − 22·37-s + 13·49-s − 10·67-s + 20·73-s − 28·97-s − 14·103-s − 38·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 23·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | + 16-s + 3.67·19-s + 3.95·31-s − 3.61·37-s + 13/7·49-s − 1.22·67-s + 2.34·73-s − 2.84·97-s − 1.37·103-s − 3.63·109-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 − 1.76·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.525257290\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.525257290\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | | \( 1 \) | |
| 7 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) | |
| 13 | $C_2^2$ | \( 1 + 23 T^{2} + p^{2} T^{4} \) | |
| good | 2 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} )( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} ) \) | 4.2.a_a_a_ae |
| 5 | $C_2^3$ | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) | 4.5.a_a_a_az |
| 11 | $C_2^3$ | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) | 4.11.a_a_a_aer |
| 17 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) | 4.17.a_abi_a_bhj |
| 19 | $C_2$$\times$$C_2^2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 - 37 T^{2} + p^{2} T^{4} ) \) | 4.19.aq_cn_lc_aenk |
| 23 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) | 4.23.a_bu_a_cjb |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.29.a_em_a_hmc |
| 31 | $C_2$$\times$$C_2^2$ | \( ( 1 - 11 T + p T^{2} )^{2}( 1 - 46 T^{2} + p^{2} T^{4} ) \) | 4.31.aw_fh_ms_ajpw |
| 37 | $C_2$$\times$$C_2^2$ | \( ( 1 + 11 T + p T^{2} )^{2}( 1 + 26 T^{2} + p^{2} T^{4} ) \) | 4.37.w_in_cbi_loi |
| 41 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) | 4.41.a_a_a_ezi |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2}( 1 + 5 T + p T^{2} )^{2} \) | 4.43.a_es_a_kzj |
| 47 | $C_2^3$ | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) | 4.47.a_a_a_adgz |
| 53 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) | 4.53.a_aec_a_mmd |
| 59 | $C_2^3$ | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) | 4.59.a_a_a_afdx |
| 61 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 121 T^{2} + p^{2} T^{4} )( 1 + 47 T^{2} + p^{2} T^{4} ) \) | 4.61.a_acw_a_cpn |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 + 5 T + p T^{2} )^{2}( 1 + 122 T^{2} + p^{2} T^{4} ) \) | 4.67.k_kv_cus_bpzk |
| 71 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) | 4.71.a_a_a_oxu |
| 73 | $C_2$$\times$$C_2^2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 143 T^{2} + p^{2} T^{4} ) \) | 4.73.au_oz_agke_cpuy |
| 79 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 142 T^{2} + p^{2} T^{4} )( 1 + 131 T^{2} + p^{2} T^{4} ) \) | 4.79.a_al_a_ajbk |
| 83 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) | 4.83.a_a_a_ujy |
| 89 | $C_2^3$ | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) | 4.89.a_a_a_alsr |
| 97 | $C_2$$\times$$C_2^2$ | \( ( 1 + 14 T + p T^{2} )^{2}( 1 + 2 T^{2} + p^{2} T^{4} ) \) | 4.97.bc_pc_ecq_bczu |
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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
−7.31152602201289277272816115250, −7.01875595559187108485832275444, −7.00616935775097049411726980193, −6.64848004922357869199928396552, −6.63348482143304098592195409213, −6.14541677967759009051957846864, −5.82774400727236184025497063293, −5.63707734713619783059223499312, −5.63460680988542412936916299648, −5.08944237764667139512425928596, −5.03629668042907450832247123833, −4.89521542358485447223477524358, −4.66109635530016156675727801898, −4.11471530981089366603726714585, −3.71342428583575967411525180307, −3.70065704819299059593613725769, −3.56098684337473417888974034631, −2.96867947641740360803436602950, −2.75428394271666342767913243927, −2.71586757530179344104755260854, −2.32384063907561017600812857739, −1.47487028388325449267476502782, −1.30097327125890924931705371390, −1.20100750757541125288506909654, −0.52069509597528120057958842358,
0.52069509597528120057958842358, 1.20100750757541125288506909654, 1.30097327125890924931705371390, 1.47487028388325449267476502782, 2.32384063907561017600812857739, 2.71586757530179344104755260854, 2.75428394271666342767913243927, 2.96867947641740360803436602950, 3.56098684337473417888974034631, 3.70065704819299059593613725769, 3.71342428583575967411525180307, 4.11471530981089366603726714585, 4.66109635530016156675727801898, 4.89521542358485447223477524358, 5.03629668042907450832247123833, 5.08944237764667139512425928596, 5.63460680988542412936916299648, 5.63707734713619783059223499312, 5.82774400727236184025497063293, 6.14541677967759009051957846864, 6.63348482143304098592195409213, 6.64848004922357869199928396552, 7.00616935775097049411726980193, 7.01875595559187108485832275444, 7.31152602201289277272816115250
