| L(s) = 1 | + 2·2-s − 3-s + 2·4-s − 2·6-s − 3·7-s + 9-s − 3·11-s − 2·12-s + 6·13-s − 6·14-s − 4·16-s − 3·17-s + 2·18-s − 19-s + 3·21-s − 6·22-s − 4·23-s + 12·26-s − 27-s − 6·28-s − 10·29-s + 2·31-s − 8·32-s + 3·33-s − 6·34-s + 2·36-s − 8·37-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.577·3-s + 4-s − 0.816·6-s − 1.13·7-s + 1/3·9-s − 0.904·11-s − 0.577·12-s + 1.66·13-s − 1.60·14-s − 16-s − 0.727·17-s + 0.471·18-s − 0.229·19-s + 0.654·21-s − 1.27·22-s − 0.834·23-s + 2.35·26-s − 0.192·27-s − 1.13·28-s − 1.85·29-s + 0.359·31-s − 1.41·32-s + 0.522·33-s − 1.02·34-s + 1/3·36-s − 1.31·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 19 | \( 1 + T \) | |
| good | 2 | \( 1 - p T + p T^{2} \) | 1.2.ac |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−9.186131792683955329903476363656, −8.317881872551815001436041820125, −7.02988413209430334324427113667, −6.35254565091942962035261866939, −5.78246884798279398607395829651, −5.02796650032629126979050185167, −3.86801042934529281313788909752, −3.42593985598594434398917386914, −2.10649008692260005676789261677, 0,
2.10649008692260005676789261677, 3.42593985598594434398917386914, 3.86801042934529281313788909752, 5.02796650032629126979050185167, 5.78246884798279398607395829651, 6.35254565091942962035261866939, 7.02988413209430334324427113667, 8.317881872551815001436041820125, 9.186131792683955329903476363656
