| L(s) = 1 | − 3-s + 4·7-s − 2·9-s − 3·11-s + 2·13-s − 17-s + 19-s − 4·21-s + 23-s + 5·27-s + 8·31-s + 3·33-s + 2·37-s − 2·39-s + 41-s − 12·43-s + 6·47-s + 9·49-s + 51-s + 4·53-s − 57-s − 12·59-s − 8·63-s − 13·67-s − 69-s + 12·71-s − 17·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.51·7-s − 2/3·9-s − 0.904·11-s + 0.554·13-s − 0.242·17-s + 0.229·19-s − 0.872·21-s + 0.208·23-s + 0.962·27-s + 1.43·31-s + 0.522·33-s + 0.328·37-s − 0.320·39-s + 0.156·41-s − 1.82·43-s + 0.875·47-s + 9/7·49-s + 0.140·51-s + 0.549·53-s − 0.132·57-s − 1.56·59-s − 1.00·63-s − 1.58·67-s − 0.120·69-s + 1.42·71-s − 1.98·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + T + p T^{2} \) | 1.17.b |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - T + p T^{2} \) | 1.41.ab |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 13 T + p T^{2} \) | 1.67.n |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 17 T + p T^{2} \) | 1.73.r |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 + 5 T + p T^{2} \) | 1.83.f |
| 89 | \( 1 - T + p T^{2} \) | 1.89.ab |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−15.14521037930557, −14.65028973674754, −14.08306997128221, −13.59016102733133, −13.19838538812401, −12.36390308866931, −11.81209897219850, −11.50792876200557, −11.00069744779490, −10.50553229948542, −10.09665677113122, −9.167751358576126, −8.472760011701286, −8.331154887678360, −7.633610741660938, −7.089141566878948, −6.201164223089835, −5.830583032470941, −5.112445880097585, −4.771655684131674, −4.171578731697036, −3.112662036226960, −2.613763438030412, −1.700961721447814, −1.020057115001984, 0,
1.020057115001984, 1.700961721447814, 2.613763438030412, 3.112662036226960, 4.171578731697036, 4.771655684131674, 5.112445880097585, 5.830583032470941, 6.201164223089835, 7.089141566878948, 7.633610741660938, 8.331154887678360, 8.472760011701286, 9.167751358576126, 10.09665677113122, 10.50553229948542, 11.00069744779490, 11.50792876200557, 11.81209897219850, 12.36390308866931, 13.19838538812401, 13.59016102733133, 14.08306997128221, 14.65028973674754, 15.14521037930557
