| L(s) = 1 | − 3-s + 5-s + 9-s − 3·11-s − 13-s − 15-s + 8·17-s + 2·19-s + 6·23-s − 4·25-s − 27-s − 29-s + 31-s + 3·33-s − 2·37-s + 39-s + 10·41-s + 45-s + 2·47-s − 8·51-s + 3·53-s − 3·55-s − 2·57-s − 9·59-s − 2·61-s − 65-s − 10·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1/3·9-s − 0.904·11-s − 0.277·13-s − 0.258·15-s + 1.94·17-s + 0.458·19-s + 1.25·23-s − 4/5·25-s − 0.192·27-s − 0.185·29-s + 0.179·31-s + 0.522·33-s − 0.328·37-s + 0.160·39-s + 1.56·41-s + 0.149·45-s + 0.291·47-s − 1.12·51-s + 0.412·53-s − 0.404·55-s − 0.264·57-s − 1.17·59-s − 0.256·61-s − 0.124·65-s − 1.22·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 244608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 244608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 2 | \( 1 \) | |
| 3 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 17 | \( 1 - 8 T + p T^{2} \) | 1.17.ai |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 - T + p T^{2} \) | 1.31.ab |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 + 9 T + p T^{2} \) | 1.59.j |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 10 T + p T^{2} \) | 1.67.k |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 + 5 T + p T^{2} \) | 1.83.f |
| 89 | \( 1 - 8 T + p T^{2} \) | 1.89.ai |
| 97 | \( 1 - 19 T + p T^{2} \) | 1.97.at |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.11461160643692, −12.56909048469976, −12.12240784941587, −11.90421055488543, −11.11190610186947, −10.84396628241760, −10.28876934772660, −9.937868094717443, −9.465544837758473, −9.077372100607585, −8.343004406931598, −7.763233810111946, −7.440233980599100, −7.137477713032844, −6.135572889006337, −6.019204753511616, −5.376141557034701, −5.090971797520493, −4.556333599861209, −3.784127877023861, −3.236658711735691, −2.731449479934721, −2.107378851269071, −1.292888106275110, −0.8791504682570172, 0,
0.8791504682570172, 1.292888106275110, 2.107378851269071, 2.731449479934721, 3.236658711735691, 3.784127877023861, 4.556333599861209, 5.090971797520493, 5.376141557034701, 6.019204753511616, 6.135572889006337, 7.137477713032844, 7.440233980599100, 7.763233810111946, 8.343004406931598, 9.077372100607585, 9.465544837758473, 9.937868094717443, 10.28876934772660, 10.84396628241760, 11.11190610186947, 11.90421055488543, 12.12240784941587, 12.56909048469976, 13.11461160643692
