| L(s) = 1 | + 3-s − 4·7-s + 9-s + 4·11-s + 6·13-s + 6·17-s + 19-s − 4·21-s + 4·23-s + 27-s + 6·29-s + 8·31-s + 4·33-s − 2·37-s + 6·39-s + 10·41-s − 8·43-s + 12·47-s + 9·49-s + 6·51-s − 2·53-s + 57-s + 4·59-s − 2·61-s − 4·63-s − 12·67-s + 4·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.51·7-s + 1/3·9-s + 1.20·11-s + 1.66·13-s + 1.45·17-s + 0.229·19-s − 0.872·21-s + 0.834·23-s + 0.192·27-s + 1.11·29-s + 1.43·31-s + 0.696·33-s − 0.328·37-s + 0.960·39-s + 1.56·41-s − 1.21·43-s + 1.75·47-s + 9/7·49-s + 0.840·51-s − 0.274·53-s + 0.132·57-s + 0.520·59-s − 0.256·61-s − 0.503·63-s − 1.46·67-s + 0.481·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.720808537\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.720808537\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 19 | \( 1 - T \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−15.53253743868194, −15.05945658032533, −14.18691843873879, −13.86760220543386, −13.52535433766754, −12.64684122988867, −12.46062323780733, −11.75379892205898, −11.13569950720741, −10.33700390159212, −9.968489097843811, −9.335082599807912, −8.905523790869199, −8.373197063754187, −7.667020227843619, −6.898699867540733, −6.385686979925208, −6.060741425411790, −5.201145755784902, −4.160207056143070, −3.695994344548922, −3.170161754411954, −2.604323778575681, −1.223560097957707, −0.9254891095120339,
0.9254891095120339, 1.223560097957707, 2.604323778575681, 3.170161754411954, 3.695994344548922, 4.160207056143070, 5.201145755784902, 6.060741425411790, 6.385686979925208, 6.898699867540733, 7.667020227843619, 8.373197063754187, 8.905523790869199, 9.335082599807912, 9.968489097843811, 10.33700390159212, 11.13569950720741, 11.75379892205898, 12.46062323780733, 12.64684122988867, 13.52535433766754, 13.86760220543386, 14.18691843873879, 15.05945658032533, 15.53253743868194
