| L(s) = 1 | − 4·7-s + 4·25-s + 32·37-s + 8·43-s − 2·49-s − 32·67-s − 16·79-s − 16·109-s + 8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4·169-s + 173-s − 16·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | − 1.51·7-s + 4/5·25-s + 5.26·37-s + 1.21·43-s − 2/7·49-s − 3.90·67-s − 1.80·79-s − 1.53·109-s + 8/11·121-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 + 4/13·169-s + 0.0760·173-s − 1.20·175-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8180684839\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8180684839\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) | |
| good | 5 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) | 4.5.a_ae_a_cc |
| 11 | $C_2^2$ | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \) | 4.11.a_ai_a_jy |
| 13 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) | 4.13.a_ae_a_ne |
| 17 | $C_2^2$ | \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \) | 4.17.a_bs_a_bow |
| 19 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) | 4.19.a_abc_a_bji |
| 23 | $C_2^2$ | \( ( 1 - 28 T^{2} + p^{2} T^{4} )^{2} \) | 4.23.a_ace_a_csw |
| 29 | $C_2^2$ | \( ( 1 - 40 T^{2} + p^{2} T^{4} )^{2} \) | 4.29.a_adc_a_ewg |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.31.a_aeu_a_inu |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) | 4.37.abg_um_aihk_cigk |
| 41 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) | 4.41.a_fk_a_mfu |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) | 4.43.ai_ho_aboy_tms |
| 47 | $C_2^2$ | \( ( 1 + 46 T^{2} + p^{2} T^{4} )^{2} \) | 4.47.a_do_a_jri |
| 53 | $C_2^2$ | \( ( 1 + 56 T^{2} + p^{2} T^{4} )^{2} \) | 4.53.a_ei_a_mys |
| 59 | $C_2^2$ | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \) | 4.59.a_afs_a_skk |
| 61 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) | 4.61.a_aca_a_mag |
| 67 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) | 4.67.bg_zc_moe_esao |
| 71 | $C_2^2$ | \( ( 1 - 124 T^{2} + p^{2} T^{4} )^{2} \) | 4.71.a_ajo_a_blre |
| 73 | $C_2^2$ | \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{2} \) | 4.73.a_ajk_a_bluk |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) | 4.79.q_pw_fzs_dafm |
| 83 | $C_2^2$ | \( ( 1 + 118 T^{2} + p^{2} T^{4} )^{2} \) | 4.83.a_jc_a_bozm |
| 89 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) | 4.89.a_fk_a_beru |
| 97 | $C_2^2$ | \( ( 1 - 170 T^{2} + p^{2} T^{4} )^{2} \) | 4.97.a_anc_a_cspi |
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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
−5.96540234377211939937171421846, −5.73780135450459054595467913766, −5.72697217453439037680699605603, −5.65707659306126034958585900375, −5.16347309132208418476105271848, −4.77299667757332560752612857223, −4.74753056007180436897972517996, −4.54207409742788470871929897061, −4.35813734184810242931397487068, −4.17876653985352456096687513964, −4.13612180376950093685972136455, −3.55293469435628186039436806496, −3.45106563058061676827104829594, −3.42996128020740950563017540543, −2.86970297117786531951481282595, −2.75331133160320226861115906339, −2.69692223347604779980805550460, −2.68001446861296984545536250745, −2.06633295405376184393002007633, −2.00318285639502746307916065769, −1.37088800537173436441269826180, −1.26073082200919827032678816460, −0.996914947118759936075404499524, −0.63082445344950819065764724194, −0.14913286812657322377418442719,
0.14913286812657322377418442719, 0.63082445344950819065764724194, 0.996914947118759936075404499524, 1.26073082200919827032678816460, 1.37088800537173436441269826180, 2.00318285639502746307916065769, 2.06633295405376184393002007633, 2.68001446861296984545536250745, 2.69692223347604779980805550460, 2.75331133160320226861115906339, 2.86970297117786531951481282595, 3.42996128020740950563017540543, 3.45106563058061676827104829594, 3.55293469435628186039436806496, 4.13612180376950093685972136455, 4.17876653985352456096687513964, 4.35813734184810242931397487068, 4.54207409742788470871929897061, 4.74753056007180436897972517996, 4.77299667757332560752612857223, 5.16347309132208418476105271848, 5.65707659306126034958585900375, 5.72697217453439037680699605603, 5.73780135450459054595467913766, 5.96540234377211939937171421846
