| L(s) = 1 | + 2-s + 2·3-s + 4-s + 5-s + 2·6-s + 8-s + 9-s + 10-s + 2·12-s − 13-s + 2·15-s + 16-s + 18-s − 2·19-s + 20-s + 6·23-s + 2·24-s + 25-s − 26-s − 4·27-s + 6·29-s + 2·30-s + 4·31-s + 32-s + 36-s + 2·37-s − 2·38-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.447·5-s + 0.816·6-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.577·12-s − 0.277·13-s + 0.516·15-s + 1/4·16-s + 0.235·18-s − 0.458·19-s + 0.223·20-s + 1.25·23-s + 0.408·24-s + 1/5·25-s − 0.196·26-s − 0.769·27-s + 1.11·29-s + 0.365·30-s + 0.718·31-s + 0.176·32-s + 1/6·36-s + 0.328·37-s − 0.324·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.524635912\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.524635912\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−8.034042417543185124679445559907, −7.30798744151240585707929637548, −6.62766586400093079500930850490, −5.87773423587761253264901175798, −5.06865919179871109007253863030, −4.34166087375820765980262818951, −3.52403007884619950350680894523, −2.68310672026322964456375987938, −2.32449601158304686237683281693, −1.08208469372552731483813194139,
1.08208469372552731483813194139, 2.32449601158304686237683281693, 2.68310672026322964456375987938, 3.52403007884619950350680894523, 4.34166087375820765980262818951, 5.06865919179871109007253863030, 5.87773423587761253264901175798, 6.62766586400093079500930850490, 7.30798744151240585707929637548, 8.034042417543185124679445559907
