| L(s) = 1 | − 5-s + 7-s + 11-s − 3·13-s + 17-s + 2·19-s + 4·23-s − 4·25-s − 2·31-s − 35-s − 7·37-s + 11·43-s − 10·47-s + 49-s − 9·53-s − 55-s + 4·59-s + 4·61-s + 3·65-s + 13·67-s − 8·71-s − 73-s + 77-s + 79-s + 9·83-s − 85-s − 3·89-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.377·7-s + 0.301·11-s − 0.832·13-s + 0.242·17-s + 0.458·19-s + 0.834·23-s − 4/5·25-s − 0.359·31-s − 0.169·35-s − 1.15·37-s + 1.67·43-s − 1.45·47-s + 1/7·49-s − 1.23·53-s − 0.134·55-s + 0.520·59-s + 0.512·61-s + 0.372·65-s + 1.58·67-s − 0.949·71-s − 0.117·73-s + 0.113·77-s + 0.112·79-s + 0.987·83-s − 0.108·85-s − 0.317·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.737280975\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.737280975\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 17 | \( 1 - T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 13 | \( 1 + 3 T + p T^{2} \) | 1.13.d |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 11 T + p T^{2} \) | 1.43.al |
| 47 | \( 1 + 10 T + p T^{2} \) | 1.47.k |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 - 13 T + p T^{2} \) | 1.67.an |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + T + p T^{2} \) | 1.73.b |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 89 | \( 1 + 3 T + p T^{2} \) | 1.89.d |
| 97 | \( 1 - 7 T + p T^{2} \) | 1.97.ah |
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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
−15.82108051335738, −15.38643000939478, −14.64678455496260, −14.35204551070594, −13.79701731404054, −12.97381119670214, −12.55926278733974, −11.92365378868471, −11.43018854335410, −10.99050200212918, −10.17976048609261, −9.697228779325314, −9.050324872734100, −8.478704456058056, −7.665965909525166, −7.429907843166951, −6.675235509865563, −5.941517989992321, −5.134033672085987, −4.757276731172088, −3.848836086985473, −3.304929695723596, −2.394881084243522, −1.590610638797747, −0.5667655103645775,
0.5667655103645775, 1.590610638797747, 2.394881084243522, 3.304929695723596, 3.848836086985473, 4.757276731172088, 5.134033672085987, 5.941517989992321, 6.675235509865563, 7.429907843166951, 7.665965909525166, 8.478704456058056, 9.050324872734100, 9.697228779325314, 10.17976048609261, 10.99050200212918, 11.43018854335410, 11.92365378868471, 12.55926278733974, 12.97381119670214, 13.79701731404054, 14.35204551070594, 14.64678455496260, 15.38643000939478, 15.82108051335738
