| L(s) = 1 | − 7-s − 4·11-s + 13-s + 4·17-s + 19-s − 4·23-s − 4·29-s − 5·31-s + 6·37-s − 12·41-s − 5·43-s + 8·47-s − 6·49-s − 12·53-s − 8·59-s + 7·61-s − 13·67-s − 12·71-s + 6·73-s + 4·77-s + 12·79-s − 8·83-s − 91-s + 13·97-s − 12·101-s + 4·103-s + 12·107-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 1.20·11-s + 0.277·13-s + 0.970·17-s + 0.229·19-s − 0.834·23-s − 0.742·29-s − 0.898·31-s + 0.986·37-s − 1.87·41-s − 0.762·43-s + 1.16·47-s − 6/7·49-s − 1.64·53-s − 1.04·59-s + 0.896·61-s − 1.58·67-s − 1.42·71-s + 0.702·73-s + 0.455·77-s + 1.35·79-s − 0.878·83-s − 0.104·91-s + 1.31·97-s − 1.19·101-s + 0.394·103-s + 1.16·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| good | 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 + 13 T + p T^{2} \) | 1.67.n |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 13 T + p T^{2} \) | 1.97.an |
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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
−8.895716002168851437744218824241, −7.929415382220924128214461090885, −7.52145300864874585303566247826, −6.39754463361252039054477440603, −5.63705675785723559506487595163, −4.89225670237683858046256523275, −3.69114066821144829351188843471, −2.92281462122494924475120025499, −1.68391378455870970336182945013, 0,
1.68391378455870970336182945013, 2.92281462122494924475120025499, 3.69114066821144829351188843471, 4.89225670237683858046256523275, 5.63705675785723559506487595163, 6.39754463361252039054477440603, 7.52145300864874585303566247826, 7.929415382220924128214461090885, 8.895716002168851437744218824241
