| L(s) = 1 | + 2-s − 2·3-s + 4-s − 2·5-s − 2·6-s − 2·7-s + 8-s + 9-s − 2·10-s − 2·12-s − 5·13-s − 2·14-s + 4·15-s + 16-s − 7·17-s + 18-s + 19-s − 2·20-s + 4·21-s − 23-s − 2·24-s − 25-s − 5·26-s + 4·27-s − 2·28-s + 8·29-s + 4·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.894·5-s − 0.816·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s − 0.632·10-s − 0.577·12-s − 1.38·13-s − 0.534·14-s + 1.03·15-s + 1/4·16-s − 1.69·17-s + 0.235·18-s + 0.229·19-s − 0.447·20-s + 0.872·21-s − 0.208·23-s − 0.408·24-s − 1/5·25-s − 0.980·26-s + 0.769·27-s − 0.377·28-s + 1.48·29-s + 0.730·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 298 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 298 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 149 | \( 1 - T \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 17 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + p T^{2} \) | 1.37.a |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 3 T + p T^{2} \) | 1.67.ad |
| 71 | \( 1 + 15 T + p T^{2} \) | 1.71.p |
| 73 | \( 1 - 9 T + p T^{2} \) | 1.73.aj |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−11.56561031246283407551316862179, −10.68472316448195504055881602591, −9.674719668106499560183943823239, −8.206606724296907088426457205693, −6.94312057827656872155076238237, −6.36416273638527124083942018068, −5.06730378076787060496001328199, −4.28596785663970327786643054301, −2.77150588602722107325738225734, 0,
2.77150588602722107325738225734, 4.28596785663970327786643054301, 5.06730378076787060496001328199, 6.36416273638527124083942018068, 6.94312057827656872155076238237, 8.206606724296907088426457205693, 9.674719668106499560183943823239, 10.68472316448195504055881602591, 11.56561031246283407551316862179
