| L(s) = 1 | + 2-s + 2·3-s + 4-s − 2·5-s + 2·6-s + 3·7-s + 8-s + 9-s − 2·10-s + 2·11-s + 2·12-s − 3·13-s + 3·14-s − 4·15-s + 16-s + 17-s + 18-s + 4·19-s − 2·20-s + 6·21-s + 2·22-s − 6·23-s + 2·24-s − 25-s − 3·26-s − 4·27-s + 3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.894·5-s + 0.816·6-s + 1.13·7-s + 0.353·8-s + 1/3·9-s − 0.632·10-s + 0.603·11-s + 0.577·12-s − 0.832·13-s + 0.801·14-s − 1.03·15-s + 1/4·16-s + 0.242·17-s + 0.235·18-s + 0.917·19-s − 0.447·20-s + 1.30·21-s + 0.426·22-s − 1.25·23-s + 0.408·24-s − 1/5·25-s − 0.588·26-s − 0.769·27-s + 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 17 | \( 1 - T \) | |
| 37 | \( 1 \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + 3 T + p T^{2} \) | 1.13.d |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + 2 T + p T^{2} \) | 1.59.c |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 + 15 T + p T^{2} \) | 1.67.p |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 83 | \( 1 + T + p T^{2} \) | 1.83.b |
| 89 | \( 1 + 8 T + p T^{2} \) | 1.89.i |
| 97 | \( 1 - 9 T + p T^{2} \) | 1.97.aj |
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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
−14.80983742288978, −14.36569890381168, −13.98913798124616, −13.55547728919348, −12.89484205016595, −12.07274604569228, −11.85442679671454, −11.55933648783900, −10.85890101408767, −10.12799704794212, −9.622949012243465, −8.981638545366686, −8.365719098709705, −7.911842538640733, −7.544048373928704, −7.155228769635615, −6.178208861813675, −5.617253728087524, −4.869002817062078, −4.325298015475808, −3.908704561230405, −3.201029429947336, −2.705546544603491, −1.886357520794618, −1.349757990022892, 0,
1.349757990022892, 1.886357520794618, 2.705546544603491, 3.201029429947336, 3.908704561230405, 4.325298015475808, 4.869002817062078, 5.617253728087524, 6.178208861813675, 7.155228769635615, 7.544048373928704, 7.911842538640733, 8.365719098709705, 8.981638545366686, 9.622949012243465, 10.12799704794212, 10.85890101408767, 11.55933648783900, 11.85442679671454, 12.07274604569228, 12.89484205016595, 13.55547728919348, 13.98913798124616, 14.36569890381168, 14.80983742288978
