| L(s) = 1 | − 3-s + 2·5-s − 4·7-s + 9-s + 2·13-s − 2·15-s + 6·17-s − 4·19-s + 4·21-s − 25-s − 27-s − 2·29-s − 4·31-s − 8·35-s − 2·37-s − 2·39-s − 2·41-s + 4·43-s + 2·45-s − 8·47-s + 9·49-s − 6·51-s + 10·53-s + 4·57-s + 4·59-s − 6·61-s − 4·63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s − 1.51·7-s + 1/3·9-s + 0.554·13-s − 0.516·15-s + 1.45·17-s − 0.917·19-s + 0.872·21-s − 1/5·25-s − 0.192·27-s − 0.371·29-s − 0.718·31-s − 1.35·35-s − 0.328·37-s − 0.320·39-s − 0.312·41-s + 0.609·43-s + 0.298·45-s − 1.16·47-s + 9/7·49-s − 0.840·51-s + 1.37·53-s + 0.529·57-s + 0.520·59-s − 0.768·61-s − 0.503·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.412785778\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.412785778\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.56561218187077, −16.04721458244726, −15.35529202273835, −14.75896139410495, −14.03223988043044, −13.42716964825037, −13.03041290940770, −12.45882165702888, −12.01020334817002, −11.13736940230884, −10.47470832316611, −10.07960502644965, −9.495236257358046, −9.064024480212361, −8.144903645098535, −7.392715529813304, −6.553606787046443, −6.275318577913717, −5.615135621960976, −5.122267211142632, −3.881872290903011, −3.495230006534216, −2.513209674368383, −1.633602583090365, −0.5626727139688470,
0.5626727139688470, 1.633602583090365, 2.513209674368383, 3.495230006534216, 3.881872290903011, 5.122267211142632, 5.615135621960976, 6.275318577913717, 6.553606787046443, 7.392715529813304, 8.144903645098535, 9.064024480212361, 9.495236257358046, 10.07960502644965, 10.47470832316611, 11.13736940230884, 12.01020334817002, 12.45882165702888, 13.03041290940770, 13.42716964825037, 14.03223988043044, 14.75896139410495, 15.35529202273835, 16.04721458244726, 16.56561218187077
