| L(s) = 1 | − 2·4-s + 3·5-s − 4·7-s − 11-s + 2·13-s + 4·16-s + 19-s − 6·20-s − 3·23-s + 4·25-s + 8·28-s + 6·29-s − 7·31-s − 12·35-s − 7·37-s − 10·43-s + 2·44-s + 9·49-s − 4·52-s − 6·53-s − 3·55-s − 3·59-s − 10·61-s − 8·64-s + 6·65-s + 11·67-s − 15·71-s + ⋯ |
| L(s) = 1 | − 4-s + 1.34·5-s − 1.51·7-s − 0.301·11-s + 0.554·13-s + 16-s + 0.229·19-s − 1.34·20-s − 0.625·23-s + 4/5·25-s + 1.51·28-s + 1.11·29-s − 1.25·31-s − 2.02·35-s − 1.15·37-s − 1.52·43-s + 0.301·44-s + 9/7·49-s − 0.554·52-s − 0.824·53-s − 0.404·55-s − 0.390·59-s − 1.28·61-s − 64-s + 0.744·65-s + 1.34·67-s − 1.78·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1881 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1881 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| 19 | \( 1 - T \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 7 T + p T^{2} \) | 1.31.h |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 11 T + p T^{2} \) | 1.67.al |
| 71 | \( 1 + 15 T + p T^{2} \) | 1.71.p |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 9 T + p T^{2} \) | 1.89.j |
| 97 | \( 1 + T + p T^{2} \) | 1.97.b |
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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
−8.994315520752837576599542028376, −8.334213787813537186877460650857, −7.10432886853576102879328057666, −6.22636985341606678010346984201, −5.73134031558754015333927704989, −4.87227236817412946007588627989, −3.68590786967197864045388603157, −2.94812083380032633044384777279, −1.58351857703013188026420779751, 0,
1.58351857703013188026420779751, 2.94812083380032633044384777279, 3.68590786967197864045388603157, 4.87227236817412946007588627989, 5.73134031558754015333927704989, 6.22636985341606678010346984201, 7.10432886853576102879328057666, 8.334213787813537186877460650857, 8.994315520752837576599542028376
