| L(s) = 1 | − 2·4-s + 5·7-s + 5·13-s + 4·16-s + 8·19-s − 5·25-s − 10·28-s − 4·31-s + 11·37-s + 8·43-s + 18·49-s − 10·52-s + 14·61-s − 8·64-s − 16·67-s − 10·73-s − 16·76-s − 4·79-s + 25·91-s − 19·97-s + 10·100-s + 101-s + 103-s + 107-s + 109-s + 20·112-s + 113-s + ⋯ |
| L(s) = 1 | − 4-s + 1.88·7-s + 1.38·13-s + 16-s + 1.83·19-s − 25-s − 1.88·28-s − 0.718·31-s + 1.80·37-s + 1.21·43-s + 18/7·49-s − 1.38·52-s + 1.79·61-s − 64-s − 1.95·67-s − 1.17·73-s − 1.83·76-s − 0.450·79-s + 2.62·91-s − 1.92·97-s + 100-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 1.88·112-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59643 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59643 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.391219146\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.391219146\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 47 | \( 1 \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 5 | \( 1 + p T^{2} \) | 1.5.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 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 11 T + p T^{2} \) | 1.37.al |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 + 16 T + p T^{2} \) | 1.67.q |
| 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 | \( 1 + 19 T + p T^{2} \) | 1.97.t |
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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
−14.25692231330310, −13.90616259126284, −13.36342011176508, −13.04110572343067, −12.19211353623122, −11.70009130333291, −11.25429302340505, −10.95330176133467, −10.14910259789718, −9.701417026205494, −8.991359721555546, −8.703271083720833, −8.097271250767126, −7.581019444855249, −7.395297881730150, −6.169534583325092, −5.592607642096180, −5.406210201607100, −4.539045192272177, −4.216261274096829, −3.616641472720886, −2.843782991119789, −1.851847506228286, −1.229679121807371, −0.7496261734760005,
0.7496261734760005, 1.229679121807371, 1.851847506228286, 2.843782991119789, 3.616641472720886, 4.216261274096829, 4.539045192272177, 5.406210201607100, 5.592607642096180, 6.169534583325092, 7.395297881730150, 7.581019444855249, 8.097271250767126, 8.703271083720833, 8.991359721555546, 9.701417026205494, 10.14910259789718, 10.95330176133467, 11.25429302340505, 11.70009130333291, 12.19211353623122, 13.04110572343067, 13.36342011176508, 13.90616259126284, 14.25692231330310
