| L(s) = 1 | + 2·5-s − 7-s − 2·11-s − 4·13-s + 2·17-s + 23-s − 25-s − 2·29-s − 2·35-s + 4·37-s − 6·41-s + 2·43-s + 4·47-s + 49-s − 4·55-s − 2·59-s + 10·61-s − 8·65-s + 2·67-s − 8·71-s − 6·73-s + 2·77-s + 4·85-s − 10·89-s + 4·91-s + 10·97-s + 101-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.377·7-s − 0.603·11-s − 1.10·13-s + 0.485·17-s + 0.208·23-s − 1/5·25-s − 0.371·29-s − 0.338·35-s + 0.657·37-s − 0.937·41-s + 0.304·43-s + 0.583·47-s + 1/7·49-s − 0.539·55-s − 0.260·59-s + 1.28·61-s − 0.992·65-s + 0.244·67-s − 0.949·71-s − 0.702·73-s + 0.227·77-s + 0.433·85-s − 1.05·89-s + 0.419·91-s + 1.01·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.674752625\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.674752625\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 23 | \( 1 - T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + 2 T + p T^{2} \) | 1.59.c |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
| 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
−13.79309536331964, −13.26423907505403, −12.99799727781098, −12.38032765984542, −11.97580506555917, −11.39899809965629, −10.74434177318161, −10.27631634875730, −9.780524791447571, −9.583990033984174, −8.929368886882648, −8.325849086662817, −7.725839596994928, −7.254323667137753, −6.747420837488484, −6.096787348458987, −5.533669768641985, −5.264954758270117, −4.530821674072086, −3.920301994207555, −3.104269683924100, −2.620806254259719, −2.076930044730166, −1.354100695016469, −0.4018262092962294,
0.4018262092962294, 1.354100695016469, 2.076930044730166, 2.620806254259719, 3.104269683924100, 3.920301994207555, 4.530821674072086, 5.264954758270117, 5.533669768641985, 6.096787348458987, 6.747420837488484, 7.254323667137753, 7.725839596994928, 8.325849086662817, 8.929368886882648, 9.583990033984174, 9.780524791447571, 10.27631634875730, 10.74434177318161, 11.39899809965629, 11.97580506555917, 12.38032765984542, 12.99799727781098, 13.26423907505403, 13.79309536331964
