| L(s) = 1 | − 2·7-s − 4·11-s − 6·13-s + 4·17-s + 19-s − 4·23-s + 6·29-s + 6·31-s + 10·37-s − 4·41-s − 12·43-s − 4·47-s − 3·49-s + 10·53-s − 10·59-s − 2·61-s − 12·67-s + 8·71-s + 2·73-s + 8·77-s − 10·79-s + 2·83-s + 8·89-s + 12·91-s − 2·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.755·7-s − 1.20·11-s − 1.66·13-s + 0.970·17-s + 0.229·19-s − 0.834·23-s + 1.11·29-s + 1.07·31-s + 1.64·37-s − 0.624·41-s − 1.82·43-s − 0.583·47-s − 3/7·49-s + 1.37·53-s − 1.30·59-s − 0.256·61-s − 1.46·67-s + 0.949·71-s + 0.234·73-s + 0.911·77-s − 1.12·79-s + 0.219·83-s + 0.847·89-s + 1.25·91-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5215377920\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5215377920\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 19 | \( 1 - T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 2 T + p T^{2} \) | 1.83.ac |
| 89 | \( 1 - 8 T + p T^{2} \) | 1.89.ai |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−12.81470049747328, −12.17844777914226, −11.96590133077552, −11.62840834825112, −10.78450516303007, −10.17951529625847, −10.10935153404312, −9.717196330396400, −9.231542271016111, −8.448345178756908, −7.941076163122460, −7.801706618341252, −7.164642995838767, −6.606649091437642, −6.208777979587564, −5.586626223678407, −5.063303713274952, −4.731716086664695, −4.146043022151250, −3.326733442912186, −2.856917338707735, −2.617086109220282, −1.854163149020454, −1.040101316394157, −0.2092214558918121,
0.2092214558918121, 1.040101316394157, 1.854163149020454, 2.617086109220282, 2.856917338707735, 3.326733442912186, 4.146043022151250, 4.731716086664695, 5.063303713274952, 5.586626223678407, 6.208777979587564, 6.606649091437642, 7.164642995838767, 7.801706618341252, 7.941076163122460, 8.448345178756908, 9.231542271016111, 9.717196330396400, 10.10935153404312, 10.17951529625847, 10.78450516303007, 11.62840834825112, 11.96590133077552, 12.17844777914226, 12.81470049747328
