| L(s) = 1 | + 2·3-s + 2·7-s + 9-s + 6·11-s + 2·13-s − 17-s + 8·19-s + 4·21-s − 6·23-s − 4·27-s + 6·29-s − 2·31-s + 12·33-s + 2·37-s + 4·39-s − 6·41-s + 4·43-s + 12·47-s − 3·49-s − 2·51-s + 6·53-s + 16·57-s − 2·61-s + 2·63-s − 8·67-s − 12·69-s + 6·71-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.755·7-s + 1/3·9-s + 1.80·11-s + 0.554·13-s − 0.242·17-s + 1.83·19-s + 0.872·21-s − 1.25·23-s − 0.769·27-s + 1.11·29-s − 0.359·31-s + 2.08·33-s + 0.328·37-s + 0.640·39-s − 0.937·41-s + 0.609·43-s + 1.75·47-s − 3/7·49-s − 0.280·51-s + 0.824·53-s + 2.11·57-s − 0.256·61-s + 0.251·63-s − 0.977·67-s − 1.44·69-s + 0.712·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.309518727\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.309518727\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 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 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 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
−15.18064267315741, −14.49116012077427, −14.23960827006188, −13.70046914885847, −13.62140417757425, −12.49228126907647, −11.93520486705124, −11.62856105680687, −11.07440119768520, −10.22882060078548, −9.661426059040218, −9.137333423389828, −8.700222024030814, −8.239632936655627, −7.553255100318958, −7.139848263895653, −6.261947494918183, −5.796178245704370, −4.930383752572318, −4.153637578357133, −3.720871249569252, −3.108912986950062, −2.259764768374674, −1.546494552108642, −0.9188498611218966,
0.9188498611218966, 1.546494552108642, 2.259764768374674, 3.108912986950062, 3.720871249569252, 4.153637578357133, 4.930383752572318, 5.796178245704370, 6.261947494918183, 7.139848263895653, 7.553255100318958, 8.239632936655627, 8.700222024030814, 9.137333423389828, 9.661426059040218, 10.22882060078548, 11.07440119768520, 11.62856105680687, 11.93520486705124, 12.49228126907647, 13.62140417757425, 13.70046914885847, 14.23960827006188, 14.49116012077427, 15.18064267315741
