| L(s) = 1 | + 3-s + 3·5-s − 7-s + 9-s − 5·11-s − 4·13-s + 3·15-s + 4·19-s − 21-s + 4·25-s + 27-s + 3·29-s + 7·31-s − 5·33-s − 3·35-s − 4·37-s − 4·39-s − 10·43-s + 3·45-s + 4·47-s + 49-s − 9·53-s − 15·55-s + 4·57-s + 5·59-s + 2·61-s − 63-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.34·5-s − 0.377·7-s + 1/3·9-s − 1.50·11-s − 1.10·13-s + 0.774·15-s + 0.917·19-s − 0.218·21-s + 4/5·25-s + 0.192·27-s + 0.557·29-s + 1.25·31-s − 0.870·33-s − 0.507·35-s − 0.657·37-s − 0.640·39-s − 1.52·43-s + 0.447·45-s + 0.583·47-s + 1/7·49-s − 1.23·53-s − 2.02·55-s + 0.529·57-s + 0.650·59-s + 0.256·61-s − 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 - T \) | |
| 7 | \( 1 + T \) | |
| 17 | \( 1 \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 - 7 T + p T^{2} \) | 1.31.ah |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 - 5 T + p T^{2} \) | 1.59.af |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + T + p T^{2} \) | 1.73.b |
| 79 | \( 1 + 5 T + p T^{2} \) | 1.79.f |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 - 4 T + p T^{2} \) | 1.89.ae |
| 97 | \( 1 + 3 T + p T^{2} \) | 1.97.d |
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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
−14.80206069767582, −14.01983944698805, −13.86485551925049, −13.37291581774781, −12.90235297942253, −12.38953613689016, −11.87463183748894, −11.10213087035273, −10.24898049999557, −10.16059498774244, −9.739362032535684, −9.190664659046894, −8.540509483339096, −7.931065757645688, −7.498851039274917, −6.779234876044064, −6.322820200520438, −5.553706489426541, −5.051137604460053, −4.751753901430521, −3.628783507515055, −2.921423166930493, −2.535829527081634, −1.991974320740382, −1.094073588616244, 0,
1.094073588616244, 1.991974320740382, 2.535829527081634, 2.921423166930493, 3.628783507515055, 4.751753901430521, 5.051137604460053, 5.553706489426541, 6.322820200520438, 6.779234876044064, 7.498851039274917, 7.931065757645688, 8.540509483339096, 9.190664659046894, 9.739362032535684, 10.16059498774244, 10.24898049999557, 11.10213087035273, 11.87463183748894, 12.38953613689016, 12.90235297942253, 13.37291581774781, 13.86485551925049, 14.01983944698805, 14.80206069767582
