| L(s) = 1 | + 4·11-s − 6·13-s + 2·17-s + 2·19-s + 23-s − 5·25-s − 6·29-s − 6·31-s − 2·37-s − 4·43-s − 2·47-s + 10·53-s − 4·59-s − 8·61-s − 12·67-s + 4·73-s − 4·79-s − 2·83-s − 10·89-s − 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 1.20·11-s − 1.66·13-s + 0.485·17-s + 0.458·19-s + 0.208·23-s − 25-s − 1.11·29-s − 1.07·31-s − 0.328·37-s − 0.609·43-s − 0.291·47-s + 1.37·53-s − 0.520·59-s − 1.02·61-s − 1.46·67-s + 0.468·73-s − 0.450·79-s − 0.219·83-s − 1.05·89-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.057711739\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.057711739\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 2 T + p T^{2} \) | 1.83.c |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 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
−13.23020839814436, −12.72599070266094, −12.25180540913722, −11.80432136244680, −11.56795969918230, −10.94919982305969, −10.25685346831139, −9.938364593176327, −9.292727916348404, −9.210183629569841, −8.506537435981208, −7.763044937007176, −7.411528661387362, −7.078727018736473, −6.449054613946514, −5.776155218764663, −5.426422337232664, −4.816534631917192, −4.230737595783088, −3.694412500005511, −3.183529765105321, −2.465537822408299, −1.796667245899120, −1.337627256678072, −0.2930137515968027,
0.2930137515968027, 1.337627256678072, 1.796667245899120, 2.465537822408299, 3.183529765105321, 3.694412500005511, 4.230737595783088, 4.816534631917192, 5.426422337232664, 5.776155218764663, 6.449054613946514, 7.078727018736473, 7.411528661387362, 7.763044937007176, 8.506537435981208, 9.210183629569841, 9.292727916348404, 9.938364593176327, 10.25685346831139, 10.94919982305969, 11.56795969918230, 11.80432136244680, 12.25180540913722, 12.72599070266094, 13.23020839814436
