| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s + 2·7-s − 8-s + 9-s + 10-s − 11-s + 12-s − 2·14-s − 15-s + 16-s + 6·17-s − 18-s − 7·19-s − 20-s + 2·21-s + 22-s − 3·23-s − 24-s + 25-s + 27-s + 2·28-s − 3·29-s + 30-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 − 0.301·11-s + 0.288·12-s − 0.534·14-s − 0.258·15-s + 1/4·16-s + 1.45·17-s − 0.235·18-s − 1.60·19-s − 0.223·20-s + 0.436·21-s + 0.213·22-s − 0.625·23-s − 0.204·24-s + 1/5·25-s + 0.192·27-s + 0.377·28-s − 0.557·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.894826098\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.894826098\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 11 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 15 T + p T^{2} \) | 1.71.ap |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 89 | \( 1 - 3 T + p T^{2} \) | 1.89.ad |
| 97 | \( 1 - 17 T + p T^{2} \) | 1.97.ar |
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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
−14.38140118952821, −14.17419792977363, −13.36136859441189, −12.76376927379553, −12.36032663883469, −11.75995845523663, −11.30415899827208, −10.69586485630702, −10.27115881058434, −9.731996522759959, −9.239195613642451, −8.387208266466651, −8.200986138089472, −7.881135374778110, −7.248469222474886, −6.571558599552428, −6.032927382580487, −5.276611542442371, −4.630071815167555, −4.020741231370263, −3.387166948293655, −2.660669960005948, −2.040119127936811, −1.370309951760724, −0.5277634725413924,
0.5277634725413924, 1.370309951760724, 2.040119127936811, 2.660669960005948, 3.387166948293655, 4.020741231370263, 4.630071815167555, 5.276611542442371, 6.032927382580487, 6.571558599552428, 7.248469222474886, 7.881135374778110, 8.200986138089472, 8.387208266466651, 9.239195613642451, 9.731996522759959, 10.27115881058434, 10.69586485630702, 11.30415899827208, 11.75995845523663, 12.36032663883469, 12.76376927379553, 13.36136859441189, 14.17419792977363, 14.38140118952821
