| L(s) = 1 | − 3-s − 3·5-s + 9-s − 11-s − 2·13-s + 3·15-s + 5·17-s − 19-s + 4·23-s + 4·25-s − 27-s + 6·29-s − 2·31-s + 33-s − 8·37-s + 2·39-s + 8·41-s + 13·43-s − 3·45-s + 13·47-s − 5·51-s + 6·53-s + 3·55-s + 57-s − 4·59-s − 13·61-s + 6·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34·5-s + 1/3·9-s − 0.301·11-s − 0.554·13-s + 0.774·15-s + 1.21·17-s − 0.229·19-s + 0.834·23-s + 4/5·25-s − 0.192·27-s + 1.11·29-s − 0.359·31-s + 0.174·33-s − 1.31·37-s + 0.320·39-s + 1.24·41-s + 1.98·43-s − 0.447·45-s + 1.89·47-s − 0.700·51-s + 0.824·53-s + 0.404·55-s + 0.132·57-s − 0.520·59-s − 1.66·61-s + 0.744·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 178752 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 178752 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 19 | \( 1 + T \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 5 T + p T^{2} \) | 1.17.af |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 - 13 T + p T^{2} \) | 1.43.an |
| 47 | \( 1 - 13 T + p T^{2} \) | 1.47.an |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 13 T + p T^{2} \) | 1.61.n |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 3 T + p T^{2} \) | 1.73.ad |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−13.35023282936959, −12.56552638853020, −12.37455753603579, −12.12133064811916, −11.63961610042408, −10.93488478347877, −10.63227141686388, −10.42547856904987, −9.420540156161928, −9.304051550397635, −8.500090526586830, −8.048138882459383, −7.536898053121980, −7.226137785572721, −6.815769865241940, −5.855648248242400, −5.730784921787234, −4.857403345649882, −4.640492752812686, −3.881657656768099, −3.569073809707693, −2.772764860141148, −2.331185887926440, −1.182281228770664, −0.7748292124539817, 0,
0.7748292124539817, 1.182281228770664, 2.331185887926440, 2.772764860141148, 3.569073809707693, 3.881657656768099, 4.640492752812686, 4.857403345649882, 5.730784921787234, 5.855648248242400, 6.815769865241940, 7.226137785572721, 7.536898053121980, 8.048138882459383, 8.500090526586830, 9.304051550397635, 9.420540156161928, 10.42547856904987, 10.63227141686388, 10.93488478347877, 11.63961610042408, 12.12133064811916, 12.37455753603579, 12.56552638853020, 13.35023282936959
