| L(s) = 1 | − 2·3-s + 5-s + 9-s − 11-s − 2·15-s − 4·17-s + 4·19-s − 6·23-s + 25-s + 4·27-s + 2·29-s + 2·33-s − 6·37-s − 10·41-s − 4·43-s + 45-s − 10·47-s − 7·49-s + 8·51-s + 2·53-s − 55-s − 8·57-s + 4·59-s − 14·61-s − 2·67-s + 12·69-s − 4·71-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.447·5-s + 1/3·9-s − 0.301·11-s − 0.516·15-s − 0.970·17-s + 0.917·19-s − 1.25·23-s + 1/5·25-s + 0.769·27-s + 0.371·29-s + 0.348·33-s − 0.986·37-s − 1.56·41-s − 0.609·43-s + 0.149·45-s − 1.45·47-s − 49-s + 1.12·51-s + 0.274·53-s − 0.134·55-s − 1.05·57-s + 0.520·59-s − 1.79·61-s − 0.244·67-s + 1.44·69-s − 0.474·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 11 | \( 1 + T \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 10 T + p T^{2} \) | 1.47.k |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−10.01203279892383742868787683670, −8.934704962347594460230473079718, −8.021503028353452900385880057808, −6.84994184953508601388768982781, −6.21828316962541960142555051462, −5.33507828467506374142167544378, −4.64900344977274697970244927698, −3.20244029150917724069310893514, −1.74060763344336841845484656009, 0,
1.74060763344336841845484656009, 3.20244029150917724069310893514, 4.64900344977274697970244927698, 5.33507828467506374142167544378, 6.21828316962541960142555051462, 6.84994184953508601388768982781, 8.021503028353452900385880057808, 8.934704962347594460230473079718, 10.01203279892383742868787683670
