| L(s) = 1 | − 3-s − 2·7-s + 9-s + 4·11-s + 2·19-s + 2·21-s − 2·23-s − 27-s + 4·29-s + 4·31-s − 4·33-s − 2·37-s + 6·41-s − 4·43-s − 8·47-s − 3·49-s + 2·53-s − 2·57-s + 4·59-s − 2·61-s − 2·63-s − 8·67-s + 2·69-s + 8·71-s − 4·73-s − 8·77-s − 8·79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.755·7-s + 1/3·9-s + 1.20·11-s + 0.458·19-s + 0.436·21-s − 0.417·23-s − 0.192·27-s + 0.742·29-s + 0.718·31-s − 0.696·33-s − 0.328·37-s + 0.937·41-s − 0.609·43-s − 1.16·47-s − 3/7·49-s + 0.274·53-s − 0.264·57-s + 0.520·59-s − 0.256·61-s − 0.251·63-s − 0.977·67-s + 0.240·69-s + 0.949·71-s − 0.468·73-s − 0.911·77-s − 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 202800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 202800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 4 T + p T^{2} \) | 1.97.ae |
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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
−13.15919584091293, −12.78418697210443, −12.31862277709845, −11.78114629945554, −11.55060694051277, −11.08249288194794, −10.32566817795691, −9.975828729447823, −9.696023800527779, −9.025470403011166, −8.670380645404623, −8.039834470790533, −7.457687500554887, −6.899967606281892, −6.495480087321351, −6.151041957114291, −5.623005756428659, −4.976066864669140, −4.411198831888298, −3.995871546942808, −3.282827294487186, −2.931514311629940, −2.032260024483121, −1.386182090329290, −0.7961515131779508, 0,
0.7961515131779508, 1.386182090329290, 2.032260024483121, 2.931514311629940, 3.282827294487186, 3.995871546942808, 4.411198831888298, 4.976066864669140, 5.623005756428659, 6.151041957114291, 6.495480087321351, 6.899967606281892, 7.457687500554887, 8.039834470790533, 8.670380645404623, 9.025470403011166, 9.696023800527779, 9.975828729447823, 10.32566817795691, 11.08249288194794, 11.55060694051277, 11.78114629945554, 12.31862277709845, 12.78418697210443, 13.15919584091293
