| L(s) = 1 | + 2-s − 2·3-s + 4-s + 5-s − 2·6-s + 2·7-s + 8-s + 9-s + 10-s + 6·11-s − 2·12-s − 2·13-s + 2·14-s − 2·15-s + 16-s − 17-s + 18-s − 8·19-s + 20-s − 4·21-s + 6·22-s + 6·23-s − 2·24-s + 25-s − 2·26-s + 4·27-s + 2·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.447·5-s − 0.816·6-s + 0.755·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.80·11-s − 0.577·12-s − 0.554·13-s + 0.534·14-s − 0.516·15-s + 1/4·16-s − 0.242·17-s + 0.235·18-s − 1.83·19-s + 0.223·20-s − 0.872·21-s + 1.27·22-s + 1.25·23-s − 0.408·24-s + 1/5·25-s − 0.392·26-s + 0.769·27-s + 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 232730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 232730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 5 | \( 1 - T \) | |
| 17 | \( 1 + T \) | |
| 37 | \( 1 \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 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
−13.01888559488164, −12.50996269492785, −12.25462088727238, −11.75453661457328, −11.42756962403170, −10.86585400757732, −10.63792993847938, −10.15412063345440, −9.379032976681532, −8.892422763782045, −8.607897832720854, −7.892009829266442, −7.138054677790729, −6.692993189801859, −6.465613397404409, −6.019494233658434, −5.338672402458562, −4.999532826058714, −4.485192631620324, −4.087484854400733, −3.493962742595839, −2.524479282791378, −2.237046851035676, −1.300608426496155, −1.041115418886348, 0,
1.041115418886348, 1.300608426496155, 2.237046851035676, 2.524479282791378, 3.493962742595839, 4.087484854400733, 4.485192631620324, 4.999532826058714, 5.338672402458562, 6.019494233658434, 6.465613397404409, 6.692993189801859, 7.138054677790729, 7.892009829266442, 8.607897832720854, 8.892422763782045, 9.379032976681532, 10.15412063345440, 10.63792993847938, 10.86585400757732, 11.42756962403170, 11.75453661457328, 12.25462088727238, 12.50996269492785, 13.01888559488164
