| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 8-s + 9-s + 10-s − 11-s + 12-s − 13-s − 15-s + 16-s + 3·17-s − 18-s − 2·19-s − 20-s + 22-s − 5·23-s − 24-s + 25-s + 26-s + 27-s + 2·29-s + 30-s − 32-s − 33-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 − 0.301·11-s + 0.288·12-s − 0.277·13-s − 0.258·15-s + 1/4·16-s + 0.727·17-s − 0.235·18-s − 0.458·19-s − 0.223·20-s + 0.213·22-s − 1.04·23-s − 0.204·24-s + 1/5·25-s + 0.196·26-s + 0.192·27-s + 0.371·29-s + 0.182·30-s − 0.176·32-s − 0.174·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| good | 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 5 T + p T^{2} \) | 1.23.f |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 7 T + p T^{2} \) | 1.37.ah |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 14 T + p T^{2} \) | 1.53.o |
| 59 | \( 1 - 11 T + p T^{2} \) | 1.59.al |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 13 T + p T^{2} \) | 1.67.n |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 13 T + p T^{2} \) | 1.73.n |
| 79 | \( 1 + 5 T + p T^{2} \) | 1.79.f |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 - 15 T + p T^{2} \) | 1.89.ap |
| 97 | \( 1 - 9 T + p T^{2} \) | 1.97.aj |
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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
−16.03741628663085, −15.63455565848094, −14.69347015994990, −14.65620701468520, −13.96524342636199, −13.14153099250438, −12.69402561982348, −12.09647469003201, −11.56888471769966, −10.85741504746511, −10.40650639753286, −9.672062340505683, −9.395443512290549, −8.563358040303806, −8.053859597738461, −7.662400422227525, −7.134867121277350, −6.213727386299018, −5.820006494371107, −4.714639835530434, −4.220947677416035, −3.319223759435720, −2.736373263645974, −1.969680594690731, −1.061058657603217, 0,
1.061058657603217, 1.969680594690731, 2.736373263645974, 3.319223759435720, 4.220947677416035, 4.714639835530434, 5.820006494371107, 6.213727386299018, 7.134867121277350, 7.662400422227525, 8.053859597738461, 8.563358040303806, 9.395443512290549, 9.672062340505683, 10.40650639753286, 10.85741504746511, 11.56888471769966, 12.09647469003201, 12.69402561982348, 13.14153099250438, 13.96524342636199, 14.65620701468520, 14.69347015994990, 15.63455565848094, 16.03741628663085
