| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 8-s + 9-s − 10-s + 12-s − 13-s + 15-s + 16-s − 6·17-s − 18-s + 19-s + 20-s − 24-s + 25-s + 26-s + 27-s − 6·29-s − 30-s − 32-s + 6·34-s + 36-s + 2·37-s − 38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.288·12-s − 0.277·13-s + 0.258·15-s + 1/4·16-s − 1.45·17-s − 0.235·18-s + 0.229·19-s + 0.223·20-s − 0.204·24-s + 1/5·25-s + 0.196·26-s + 0.192·27-s − 1.11·29-s − 0.182·30-s − 0.176·32-s + 1.02·34-s + 1/6·36-s + 0.328·37-s − 0.162·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363090 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363090 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 3 | \( 1 - T \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| 19 | \( 1 - T \) | |
| good | 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−12.66605646163507, −12.54097088057579, −11.57446996072613, −11.30940527567775, −10.96476628397168, −10.37876158200371, −9.892087711353395, −9.520031463413106, −9.082841491152032, −8.787874580372732, −8.247585839276306, −7.688988854833357, −7.383188725000644, −6.702551208842363, −6.548404283833427, −5.692920191342706, −5.473298603172223, −4.659220600476985, −4.166688437961174, −3.702727837310284, −2.951488023956189, −2.336618138970025, −2.217625730025730, −1.425850972461958, −0.7863283539809420, 0,
0.7863283539809420, 1.425850972461958, 2.217625730025730, 2.336618138970025, 2.951488023956189, 3.702727837310284, 4.166688437961174, 4.659220600476985, 5.473298603172223, 5.692920191342706, 6.548404283833427, 6.702551208842363, 7.383188725000644, 7.688988854833357, 8.247585839276306, 8.787874580372732, 9.082841491152032, 9.520031463413106, 9.892087711353395, 10.37876158200371, 10.96476628397168, 11.30940527567775, 11.57446996072613, 12.54097088057579, 12.66605646163507
