| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s + 2·7-s − 8-s + 9-s − 10-s + 2·11-s + 12-s + 6·13-s − 2·14-s + 15-s + 16-s + 3·17-s − 18-s − 2·19-s + 20-s + 2·21-s − 2·22-s + 8·23-s − 24-s − 4·25-s − 6·26-s + 27-s + 2·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.603·11-s + 0.288·12-s + 1.66·13-s − 0.534·14-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 + 0.436·21-s − 0.426·22-s + 1.66·23-s − 0.204·24-s − 4/5·25-s − 1.17·26-s + 0.192·27-s + 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8214 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8214 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.929716290\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.929716290\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 37 | \( 1 \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - T + p T^{2} \) | 1.29.ab |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + T + p T^{2} \) | 1.89.b |
| 97 | \( 1 + 17 T + p T^{2} \) | 1.97.r |
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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
−8.028459398868020788379059150600, −7.22503374907782683641840391310, −6.56253958267682466349663608224, −5.87811392054043406977921478478, −5.11149866059349047022350629927, −4.08646486221771658950230828413, −3.43116533216982878101391391651, −2.50604390489952377620560351718, −1.53119748215332517572066201466, −1.05283078710042221743579867198,
1.05283078710042221743579867198, 1.53119748215332517572066201466, 2.50604390489952377620560351718, 3.43116533216982878101391391651, 4.08646486221771658950230828413, 5.11149866059349047022350629927, 5.87811392054043406977921478478, 6.56253958267682466349663608224, 7.22503374907782683641840391310, 8.028459398868020788379059150600
