| L(s) = 1 | − 2·11-s − 2·13-s − 4·17-s − 19-s − 2·23-s − 6·29-s + 4·31-s + 2·37-s − 2·41-s + 4·43-s − 10·47-s − 7·49-s + 6·53-s − 4·59-s − 2·61-s + 4·67-s + 4·71-s + 6·73-s + 8·79-s − 2·83-s + 6·89-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 0.603·11-s − 0.554·13-s − 0.970·17-s − 0.229·19-s − 0.417·23-s − 1.11·29-s + 0.718·31-s + 0.328·37-s − 0.312·41-s + 0.609·43-s − 1.45·47-s − 49-s + 0.824·53-s − 0.520·59-s − 0.256·61-s + 0.488·67-s + 0.474·71-s + 0.702·73-s + 0.900·79-s − 0.219·83-s + 0.635·89-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 19 | \( 1 + T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 10 T + p T^{2} \) | 1.47.k |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 2 T + p T^{2} \) | 1.83.c |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−13.03925488067995, −12.66488030198541, −12.00910131411266, −11.65479281476069, −11.11589039967360, −10.76940593381190, −10.22859714309146, −9.752631664382052, −9.370727707662900, −8.836133770669684, −8.314536370538142, −7.804468719989889, −7.540340322923525, −6.767339311728328, −6.481999824446566, −5.932579972672235, −5.281036141500595, −4.887448508103858, −4.406519790626509, −3.797319823644512, −3.255019403057945, −2.561588326736553, −2.149020745249891, −1.582570296851180, −0.6252538672412759, 0,
0.6252538672412759, 1.582570296851180, 2.149020745249891, 2.561588326736553, 3.255019403057945, 3.797319823644512, 4.406519790626509, 4.887448508103858, 5.281036141500595, 5.932579972672235, 6.481999824446566, 6.767339311728328, 7.540340322923525, 7.804468719989889, 8.314536370538142, 8.836133770669684, 9.370727707662900, 9.752631664382052, 10.22859714309146, 10.76940593381190, 11.11589039967360, 11.65479281476069, 12.00910131411266, 12.66488030198541, 13.03925488067995
