| L(s) = 1 | − 3-s + 5-s + 2·7-s + 9-s − 2·11-s − 15-s − 2·17-s − 2·19-s − 2·21-s + 8·23-s + 25-s − 27-s + 6·29-s − 2·31-s + 2·33-s + 2·35-s + 2·37-s + 2·41-s + 45-s + 6·47-s − 3·49-s + 2·51-s + 10·53-s − 2·55-s + 2·57-s + 14·59-s + 10·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s − 0.603·11-s − 0.258·15-s − 0.485·17-s − 0.458·19-s − 0.436·21-s + 1.66·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s − 0.359·31-s + 0.348·33-s + 0.338·35-s + 0.328·37-s + 0.312·41-s + 0.149·45-s + 0.875·47-s − 3/7·49-s + 0.280·51-s + 1.37·53-s − 0.269·55-s + 0.264·57-s + 1.82·59-s + 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.072292384\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.072292384\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 - 14 T + p T^{2} \) | 1.59.ao |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 18 T + p T^{2} \) | 1.89.as |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−13.28994213946299, −12.82458364725488, −12.37639625916588, −11.64222753924767, −11.45641730681732, −10.84281778578131, −10.43876580860978, −10.17608985437173, −9.380128365817544, −8.951131936334691, −8.465929206734093, −8.003226629966189, −7.278086835493231, −6.958792665600879, −6.422451070718855, −5.786098074850710, −5.327034079403629, −4.873791175306280, −4.463906989125813, −3.791053310485658, −3.011120832793933, −2.370642029049020, −1.953548827715035, −1.009744432384274, −0.6304116383648531,
0.6304116383648531, 1.009744432384274, 1.953548827715035, 2.370642029049020, 3.011120832793933, 3.791053310485658, 4.463906989125813, 4.873791175306280, 5.327034079403629, 5.786098074850710, 6.422451070718855, 6.958792665600879, 7.278086835493231, 8.003226629966189, 8.465929206734093, 8.951131936334691, 9.380128365817544, 10.17608985437173, 10.43876580860978, 10.84281778578131, 11.45641730681732, 11.64222753924767, 12.37639625916588, 12.82458364725488, 13.28994213946299
