| L(s) = 1 | − 2·3-s − 7-s + 9-s + 6·11-s − 2·13-s + 17-s + 2·21-s + 4·23-s + 4·27-s − 8·29-s − 12·33-s + 4·37-s + 4·39-s − 2·41-s − 8·43-s + 8·47-s + 49-s − 2·51-s − 6·53-s + 4·59-s + 8·61-s − 63-s − 16·67-s − 8·69-s + 4·71-s − 10·73-s − 6·77-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.377·7-s + 1/3·9-s + 1.80·11-s − 0.554·13-s + 0.242·17-s + 0.436·21-s + 0.834·23-s + 0.769·27-s − 1.48·29-s − 2.08·33-s + 0.657·37-s + 0.640·39-s − 0.312·41-s − 1.21·43-s + 1.16·47-s + 1/7·49-s − 0.280·51-s − 0.824·53-s + 0.520·59-s + 1.02·61-s − 0.125·63-s − 1.95·67-s − 0.963·69-s + 0.474·71-s − 1.17·73-s − 0.683·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 190400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.211271832\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.211271832\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 17 | \( 1 - T \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 + 16 T + p T^{2} \) | 1.67.q |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−13.06275026873111, −12.48942079902995, −12.00759307771164, −11.78051520753718, −11.26863171572441, −10.95501585967145, −10.31072455815531, −9.790942408363853, −9.432960447975881, −8.834215021880555, −8.568327479189279, −7.602415664874581, −7.212768033011145, −6.756419708239866, −6.296029075029120, −5.814368080218066, −5.445890648297863, −4.683267059436225, −4.403182240633639, −3.590344684394710, −3.268610394122801, −2.406582473006879, −1.639030663421483, −1.069298032407668, −0.3859708538763104,
0.3859708538763104, 1.069298032407668, 1.639030663421483, 2.406582473006879, 3.268610394122801, 3.590344684394710, 4.403182240633639, 4.683267059436225, 5.445890648297863, 5.814368080218066, 6.296029075029120, 6.756419708239866, 7.212768033011145, 7.602415664874581, 8.568327479189279, 8.834215021880555, 9.432960447975881, 9.790942408363853, 10.31072455815531, 10.95501585967145, 11.26863171572441, 11.78051520753718, 12.00759307771164, 12.48942079902995, 13.06275026873111
