| L(s) = 1 | + 3-s + 5-s − 2·7-s + 9-s + 2·11-s + 4·13-s + 15-s + 2·17-s + 19-s − 2·21-s − 4·23-s + 25-s + 27-s − 4·29-s + 2·33-s − 2·35-s + 4·39-s + 10·43-s + 45-s + 12·47-s − 3·49-s + 2·51-s + 2·53-s + 2·55-s + 57-s − 4·59-s − 2·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.755·7-s + 1/3·9-s + 0.603·11-s + 1.10·13-s + 0.258·15-s + 0.485·17-s + 0.229·19-s − 0.436·21-s − 0.834·23-s + 1/5·25-s + 0.192·27-s − 0.742·29-s + 0.348·33-s − 0.338·35-s + 0.640·39-s + 1.52·43-s + 0.149·45-s + 1.75·47-s − 3/7·49-s + 0.280·51-s + 0.274·53-s + 0.269·55-s + 0.132·57-s − 0.520·59-s − 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.262595219\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.262595219\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 19 | \( 1 - T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + p T^{2} \) | 1.37.a |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 16 T + p T^{2} \) | 1.67.aq |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 16 T + p T^{2} \) | 1.97.q |
| 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
−15.73518697300009, −15.36913737564800, −14.44135435712406, −14.23373186114744, −13.52368046078727, −13.26121139540754, −12.39244100354487, −12.19913902904150, −11.20474978090964, −10.77642048913624, −10.05700441747169, −9.463295539631887, −9.185964860322146, −8.458518573394725, −7.880918150380375, −7.146593574188032, −6.565517126257919, −5.875547358161300, −5.529750653314712, −4.323217502892525, −3.834344614271950, −3.226925497717003, −2.432625847833450, −1.610992132716741, −0.7625019381352025,
0.7625019381352025, 1.610992132716741, 2.432625847833450, 3.226925497717003, 3.834344614271950, 4.323217502892525, 5.529750653314712, 5.875547358161300, 6.565517126257919, 7.146593574188032, 7.880918150380375, 8.458518573394725, 9.185964860322146, 9.463295539631887, 10.05700441747169, 10.77642048913624, 11.20474978090964, 12.19913902904150, 12.39244100354487, 13.26121139540754, 13.52368046078727, 14.23373186114744, 14.44135435712406, 15.36913737564800, 15.73518697300009
