| L(s) = 1 | − 2·4-s + 5·7-s + 5·13-s + 4·16-s − 19-s − 10·28-s − 7·31-s − 10·37-s + 5·43-s + 18·49-s − 10·52-s − 13·61-s − 8·64-s + 5·67-s − 10·73-s + 2·76-s − 4·79-s + 25·91-s + 5·97-s + 20·103-s − 19·109-s + 20·112-s + ⋯ |
| L(s) = 1 | − 4-s + 1.88·7-s + 1.38·13-s + 16-s − 0.229·19-s − 1.88·28-s − 1.25·31-s − 1.64·37-s + 0.762·43-s + 18/7·49-s − 1.38·52-s − 1.66·61-s − 64-s + 0.610·67-s − 1.17·73-s + 0.229·76-s − 0.450·79-s + 2.62·91-s + 0.507·97-s + 1.97·103-s − 1.81·109-s + 1.88·112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.208363907\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.208363907\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 7 | \( 1 - 5 T + p T^{2} \) | 1.7.af |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 7 T + p T^{2} \) | 1.31.h |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 5 T + p T^{2} \) | 1.43.af |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 13 T + p T^{2} \) | 1.61.n |
| 67 | \( 1 - 5 T + p T^{2} \) | 1.67.af |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 5 T + p T^{2} \) | 1.97.af |
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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
−12.23814096251703440015939659085, −11.17123610633436781993835627791, −10.50317121206251201104711594578, −8.995186005196822049947024673068, −8.459962290175805266187907432817, −7.51612763708038261050643932552, −5.77556571014558049232659651532, −4.83022728068118968978622921639, −3.80273248963840130172861112356, −1.51470298744943699034589542418,
1.51470298744943699034589542418, 3.80273248963840130172861112356, 4.83022728068118968978622921639, 5.77556571014558049232659651532, 7.51612763708038261050643932552, 8.459962290175805266187907432817, 8.995186005196822049947024673068, 10.50317121206251201104711594578, 11.17123610633436781993835627791, 12.23814096251703440015939659085
