| L(s) = 1 | − 3-s + 9-s − 4·11-s − 13-s + 2·17-s + 4·19-s + 4·23-s − 27-s − 6·29-s + 4·31-s + 4·33-s − 6·37-s + 39-s − 2·41-s + 12·43-s + 8·47-s − 7·49-s − 2·51-s − 14·53-s − 4·57-s − 12·59-s − 2·61-s − 4·69-s − 8·71-s + 2·73-s − 8·79-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 1.20·11-s − 0.277·13-s + 0.485·17-s + 0.917·19-s + 0.834·23-s − 0.192·27-s − 1.11·29-s + 0.718·31-s + 0.696·33-s − 0.986·37-s + 0.160·39-s − 0.312·41-s + 1.82·43-s + 1.16·47-s − 49-s − 0.280·51-s − 1.92·53-s − 0.529·57-s − 1.56·59-s − 0.256·61-s − 0.481·69-s − 0.949·71-s + 0.234·73-s − 0.900·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 14 T + p T^{2} \) | 1.53.o |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−16.20722608266443, −15.64048628232474, −15.48929122441083, −14.54308614801571, −14.10878283087719, −13.36460122288349, −12.95034844595814, −12.29260667703303, −11.91854557854786, −11.02260454181626, −10.80056288064979, −10.11722146574502, −9.490927836523903, −8.988484237455616, −8.043461710467623, −7.523184934794751, −7.167707192508137, −6.147384622318813, −5.713107619379816, −4.971298191813751, −4.620710438495678, −3.469963056919942, −2.944205321157291, −1.992340286234635, −1.017782068149340, 0,
1.017782068149340, 1.992340286234635, 2.944205321157291, 3.469963056919942, 4.620710438495678, 4.971298191813751, 5.713107619379816, 6.147384622318813, 7.167707192508137, 7.523184934794751, 8.043461710467623, 8.988484237455616, 9.490927836523903, 10.11722146574502, 10.80056288064979, 11.02260454181626, 11.91854557854786, 12.29260667703303, 12.95034844595814, 13.36460122288349, 14.10878283087719, 14.54308614801571, 15.48929122441083, 15.64048628232474, 16.20722608266443
