| L(s) = 1 | − 3-s − 2·7-s − 2·9-s − 3·11-s + 4·13-s − 5·19-s + 2·21-s − 6·23-s + 5·27-s + 2·31-s + 3·33-s + 2·37-s − 4·39-s + 3·41-s − 4·43-s + 12·47-s − 3·49-s − 6·53-s + 5·57-s − 2·61-s + 4·63-s − 13·67-s + 6·69-s + 12·71-s + 11·73-s + 6·77-s − 10·79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.755·7-s − 2/3·9-s − 0.904·11-s + 1.10·13-s − 1.14·19-s + 0.436·21-s − 1.25·23-s + 0.962·27-s + 0.359·31-s + 0.522·33-s + 0.328·37-s − 0.640·39-s + 0.468·41-s − 0.609·43-s + 1.75·47-s − 3/7·49-s − 0.824·53-s + 0.662·57-s − 0.256·61-s + 0.503·63-s − 1.58·67-s + 0.722·69-s + 1.42·71-s + 1.28·73-s + 0.683·77-s − 1.12·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4332029087\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4332029087\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 17 | \( 1 \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 13 T + p T^{2} \) | 1.67.n |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 - 15 T + p T^{2} \) | 1.89.ap |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−13.62251186699238, −13.09197780833849, −12.58013787464619, −12.22516023824843, −11.67408672353998, −11.02264989783459, −10.78942211840332, −10.32279882472258, −9.769019362367446, −9.202016026063305, −8.579788512970620, −8.240669708443957, −7.745100831279262, −7.004708412941053, −6.329471227420968, −6.122433519132340, −5.691666506715997, −5.037818617043892, −4.378745398173701, −3.838218913148084, −3.195551786653421, −2.605814095563707, −2.027984627614279, −1.080607982074925, −0.2289362176205603,
0.2289362176205603, 1.080607982074925, 2.027984627614279, 2.605814095563707, 3.195551786653421, 3.838218913148084, 4.378745398173701, 5.037818617043892, 5.691666506715997, 6.122433519132340, 6.329471227420968, 7.004708412941053, 7.745100831279262, 8.240669708443957, 8.579788512970620, 9.202016026063305, 9.769019362367446, 10.32279882472258, 10.78942211840332, 11.02264989783459, 11.67408672353998, 12.22516023824843, 12.58013787464619, 13.09197780833849, 13.62251186699238
