| L(s) = 1 | − 5-s + 7-s − 2·11-s + 3·17-s + 19-s + 4·23-s − 4·25-s + 29-s + 4·31-s − 35-s − 3·37-s + 7·41-s − 9·43-s + 47-s − 6·49-s − 2·53-s + 2·55-s − 3·59-s − 6·61-s − 12·67-s − 16·71-s − 10·73-s − 2·77-s + 10·79-s − 3·85-s − 6·89-s − 95-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.377·7-s − 0.603·11-s + 0.727·17-s + 0.229·19-s + 0.834·23-s − 4/5·25-s + 0.185·29-s + 0.718·31-s − 0.169·35-s − 0.493·37-s + 1.09·41-s − 1.37·43-s + 0.145·47-s − 6/7·49-s − 0.274·53-s + 0.269·55-s − 0.390·59-s − 0.768·61-s − 1.46·67-s − 1.89·71-s − 1.17·73-s − 0.227·77-s + 1.12·79-s − 0.325·85-s − 0.635·89-s − 0.102·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 29 | \( 1 - T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 - 7 T + p T^{2} \) | 1.41.ah |
| 43 | \( 1 + 9 T + p T^{2} \) | 1.43.j |
| 47 | \( 1 - T + p T^{2} \) | 1.47.ab |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 16 T + p T^{2} \) | 1.71.q |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + p T^{2} \) | 1.97.a |
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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
−16.17957845132688, −15.47226314190743, −15.28069636479297, −14.44274441251394, −14.13051336810200, −13.24457510679568, −13.04711853535023, −12.04266444998118, −11.88397302745961, −11.17813612569588, −10.55677491534673, −10.06765306099521, −9.403464068116130, −8.692633651448434, −8.136372399607573, −7.550468288015790, −7.165933846203757, −6.167004381071266, −5.703907549026153, −4.790709965475142, −4.510285399183342, −3.392256148267414, −3.016487489881610, −1.968181428827693, −1.116618148775415, 0,
1.116618148775415, 1.968181428827693, 3.016487489881610, 3.392256148267414, 4.510285399183342, 4.790709965475142, 5.703907549026153, 6.167004381071266, 7.165933846203757, 7.550468288015790, 8.136372399607573, 8.692633651448434, 9.403464068116130, 10.06765306099521, 10.55677491534673, 11.17813612569588, 11.88397302745961, 12.04266444998118, 13.04711853535023, 13.24457510679568, 14.13051336810200, 14.44274441251394, 15.28069636479297, 15.47226314190743, 16.17957845132688
