| L(s) = 1 | − 3·7-s + 3·11-s − 2·17-s − 2·19-s + 6·23-s − 5·25-s + 2·29-s − 4·31-s + 37-s − 7·41-s + 4·43-s − 47-s + 2·49-s − 9·53-s − 8·59-s − 4·61-s + 12·67-s + 5·71-s − 13·73-s − 9·77-s − 10·79-s + 83-s + 2·89-s − 12·97-s + 9·101-s − 8·103-s − 12·107-s + ⋯ |
| L(s) = 1 | − 1.13·7-s + 0.904·11-s − 0.485·17-s − 0.458·19-s + 1.25·23-s − 25-s + 0.371·29-s − 0.718·31-s + 0.164·37-s − 1.09·41-s + 0.609·43-s − 0.145·47-s + 2/7·49-s − 1.23·53-s − 1.04·59-s − 0.512·61-s + 1.46·67-s + 0.593·71-s − 1.52·73-s − 1.02·77-s − 1.12·79-s + 0.109·83-s + 0.211·89-s − 1.21·97-s + 0.895·101-s − 0.788·103-s − 1.16·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 37 | \( 1 - T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 41 | \( 1 + 7 T + p T^{2} \) | 1.41.h |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + T + p T^{2} \) | 1.47.b |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 5 T + p T^{2} \) | 1.71.af |
| 73 | \( 1 + 13 T + p T^{2} \) | 1.73.n |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - T + p T^{2} \) | 1.83.ab |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 + 12 T + p T^{2} \) | 1.97.m |
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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
−8.637798462537089082559029210080, −7.65286120224110903388505491453, −6.72433556984559570880095854341, −6.40256795051194833967277103826, −5.43110991062316119422108197791, −4.38932651849542855280266113316, −3.59781441472728197557363900163, −2.76426174907496236443941498069, −1.50974967736814861270341338685, 0,
1.50974967736814861270341338685, 2.76426174907496236443941498069, 3.59781441472728197557363900163, 4.38932651849542855280266113316, 5.43110991062316119422108197791, 6.40256795051194833967277103826, 6.72433556984559570880095854341, 7.65286120224110903388505491453, 8.637798462537089082559029210080
