| L(s) = 1 | − 3-s + 7-s + 9-s − 2·11-s + 4·13-s − 17-s − 19-s − 21-s − 6·23-s − 27-s + 2·29-s − 2·31-s + 2·33-s + 2·37-s − 4·39-s + 12·41-s + 10·43-s − 11·47-s − 6·49-s + 51-s + 3·53-s + 57-s + 4·59-s − 10·61-s + 63-s − 7·67-s + 6·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.603·11-s + 1.10·13-s − 0.242·17-s − 0.229·19-s − 0.218·21-s − 1.25·23-s − 0.192·27-s + 0.371·29-s − 0.359·31-s + 0.348·33-s + 0.328·37-s − 0.640·39-s + 1.87·41-s + 1.52·43-s − 1.60·47-s − 6/7·49-s + 0.140·51-s + 0.412·53-s + 0.132·57-s + 0.520·59-s − 1.28·61-s + 0.125·63-s − 0.855·67-s + 0.722·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 387600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| 19 | \( 1 + T \) | |
| good | 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 12 T + p T^{2} \) | 1.41.am |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 + 11 T + p T^{2} \) | 1.47.l |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 7 T + p T^{2} \) | 1.67.h |
| 71 | \( 1 - 9 T + p T^{2} \) | 1.71.aj |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + T + p T^{2} \) | 1.83.b |
| 89 | \( 1 - 15 T + p T^{2} \) | 1.89.ap |
| 97 | \( 1 - 7 T + p T^{2} \) | 1.97.ah |
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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
−12.75480049525059, −12.18736107824230, −11.75901488733178, −11.18206540256524, −11.02042736627778, −10.47356153574188, −10.18505316726087, −9.396742125011408, −9.233202464955910, −8.482904078286181, −8.119945653973389, −7.656888861185297, −7.313465481694478, −6.471305686691816, −6.159532659364787, −5.891329479991341, −5.230324759403205, −4.727752722963463, −4.272973524871612, −3.782323945183516, −3.218889769090952, −2.456617238638385, −2.020075271998112, −1.328022748663151, −0.7422664090283648, 0,
0.7422664090283648, 1.328022748663151, 2.020075271998112, 2.456617238638385, 3.218889769090952, 3.782323945183516, 4.272973524871612, 4.727752722963463, 5.230324759403205, 5.891329479991341, 6.159532659364787, 6.471305686691816, 7.313465481694478, 7.656888861185297, 8.119945653973389, 8.482904078286181, 9.233202464955910, 9.396742125011408, 10.18505316726087, 10.47356153574188, 11.02042736627778, 11.18206540256524, 11.75901488733178, 12.18736107824230, 12.75480049525059
