| L(s) = 1 | + 2·3-s − 2·5-s + 9-s − 4·15-s − 2·17-s − 4·19-s + 23-s − 25-s − 4·27-s + 10·29-s − 6·31-s − 2·37-s + 4·43-s − 2·45-s + 6·47-s − 4·51-s + 2·53-s − 8·57-s − 2·59-s + 2·61-s + 2·69-s − 8·71-s + 4·73-s − 2·75-s + 4·79-s − 11·81-s + 4·85-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s + 1/3·9-s − 1.03·15-s − 0.485·17-s − 0.917·19-s + 0.208·23-s − 1/5·25-s − 0.769·27-s + 1.85·29-s − 1.07·31-s − 0.328·37-s + 0.609·43-s − 0.298·45-s + 0.875·47-s − 0.560·51-s + 0.274·53-s − 1.05·57-s − 0.260·59-s + 0.256·61-s + 0.240·69-s − 0.949·71-s + 0.468·73-s − 0.230·75-s + 0.450·79-s − 1.22·81-s + 0.433·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72128 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 2 T + p T^{2} \) | 1.59.c |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.28756017264667, −13.98925692874513, −13.45866889053865, −12.82848854513954, −12.51499355455465, −11.75520803294755, −11.52169677522719, −10.66983225483350, −10.47936024046684, −9.687727980922233, −9.034911122457970, −8.776462914444915, −8.264294422387110, −7.799553491637008, −7.309627770374842, −6.748343668949878, −6.105768459535203, −5.452014282810387, −4.560635708783658, −4.268701579758384, −3.563586929298755, −3.147574708738588, −2.377366972538770, −1.967013776223835, −0.8868609574556271, 0,
0.8868609574556271, 1.967013776223835, 2.377366972538770, 3.147574708738588, 3.563586929298755, 4.268701579758384, 4.560635708783658, 5.452014282810387, 6.105768459535203, 6.748343668949878, 7.309627770374842, 7.799553491637008, 8.264294422387110, 8.776462914444915, 9.034911122457970, 9.687727980922233, 10.47936024046684, 10.66983225483350, 11.52169677522719, 11.75520803294755, 12.51499355455465, 12.82848854513954, 13.45866889053865, 13.98925692874513, 14.28756017264667
