| L(s) = 1 | − 2·3-s − 5-s + 2·7-s + 9-s + 11-s − 2·13-s + 2·15-s − 2·17-s − 4·21-s + 2·23-s + 25-s + 4·27-s − 2·29-s + 4·31-s − 2·33-s − 2·35-s − 6·37-s + 4·39-s + 6·41-s − 6·43-s − 45-s + 6·47-s − 3·49-s + 4·51-s − 6·53-s − 55-s − 6·61-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s + 0.301·11-s − 0.554·13-s + 0.516·15-s − 0.485·17-s − 0.872·21-s + 0.417·23-s + 1/5·25-s + 0.769·27-s − 0.371·29-s + 0.718·31-s − 0.348·33-s − 0.338·35-s − 0.986·37-s + 0.640·39-s + 0.937·41-s − 0.914·43-s − 0.149·45-s + 0.875·47-s − 3/7·49-s + 0.560·51-s − 0.824·53-s − 0.134·55-s − 0.768·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 11 | \( 1 - T \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−8.843984192067285101836130819770, −8.088328180965082136300526227350, −7.18926419098018223210703504685, −6.50409204323732348541055422680, −5.55932699453271263805116094930, −4.86901233271988171005468978851, −4.16042505835541806862747528745, −2.81577938725062240394015803216, −1.40075843522825882104989949861, 0,
1.40075843522825882104989949861, 2.81577938725062240394015803216, 4.16042505835541806862747528745, 4.86901233271988171005468978851, 5.55932699453271263805116094930, 6.50409204323732348541055422680, 7.18926419098018223210703504685, 8.088328180965082136300526227350, 8.843984192067285101836130819770
