| L(s) = 1 | − 3-s + 2·7-s + 9-s + 6·13-s − 2·21-s − 6·23-s − 27-s − 6·29-s + 10·31-s + 2·37-s − 6·39-s − 4·43-s + 8·47-s − 3·49-s − 6·53-s − 10·61-s + 2·63-s + 8·67-s + 6·69-s + 10·71-s − 16·73-s − 6·79-s + 81-s + 16·83-s + 6·87-s + 10·89-s + 12·91-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.755·7-s + 1/3·9-s + 1.66·13-s − 0.436·21-s − 1.25·23-s − 0.192·27-s − 1.11·29-s + 1.79·31-s + 0.328·37-s − 0.960·39-s − 0.609·43-s + 1.16·47-s − 3/7·49-s − 0.824·53-s − 1.28·61-s + 0.251·63-s + 0.977·67-s + 0.722·69-s + 1.18·71-s − 1.87·73-s − 0.675·79-s + 1/9·81-s + 1.75·83-s + 0.643·87-s + 1.05·89-s + 1.25·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 346800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 346800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.326676685\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.326676685\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 17 | \( 1 \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 + 16 T + p T^{2} \) | 1.73.q |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
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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
−12.39205749332233, −12.02141725243228, −11.63387081296042, −11.22388489470906, −10.74322420573001, −10.50453242390447, −9.878715280685907, −9.383231523090806, −8.906377173379826, −8.312593575037965, −7.945674728637867, −7.698091671189950, −6.854350495161620, −6.386173515160358, −6.102399109635174, −5.543478024400554, −5.116400689544692, −4.397349514870138, −4.141805860782095, −3.557254963313489, −2.948767114582677, −2.173335501515473, −1.602039518059283, −1.163634993839782, −0.4393577685801234,
0.4393577685801234, 1.163634993839782, 1.602039518059283, 2.173335501515473, 2.948767114582677, 3.557254963313489, 4.141805860782095, 4.397349514870138, 5.116400689544692, 5.543478024400554, 6.102399109635174, 6.386173515160358, 6.854350495161620, 7.698091671189950, 7.945674728637867, 8.312593575037965, 8.906377173379826, 9.383231523090806, 9.878715280685907, 10.50453242390447, 10.74322420573001, 11.22388489470906, 11.63387081296042, 12.02141725243228, 12.39205749332233
