| L(s) = 1 | + 4·7-s + 11-s + 4·13-s + 6·17-s − 4·19-s − 5·25-s − 29-s + 4·31-s − 2·37-s − 6·41-s + 8·43-s + 6·47-s + 9·49-s + 6·59-s − 2·61-s − 4·67-s − 4·73-s + 4·77-s + 10·79-s − 12·83-s − 12·89-s + 16·91-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 1.51·7-s + 0.301·11-s + 1.10·13-s + 1.45·17-s − 0.917·19-s − 25-s − 0.185·29-s + 0.718·31-s − 0.328·37-s − 0.937·41-s + 1.21·43-s + 0.875·47-s + 9/7·49-s + 0.781·59-s − 0.256·61-s − 0.488·67-s − 0.468·73-s + 0.455·77-s + 1.12·79-s − 1.31·83-s − 1.27·89-s + 1.67·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 183744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 183744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.139503999\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.139503999\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| 29 | \( 1 + T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−13.17923286524785, −12.61767903361409, −12.02969207334776, −11.82158383289109, −11.19273387294886, −10.90734701469700, −10.36015577749858, −9.923676929672644, −9.308064489001227, −8.675523425425961, −8.320974465989526, −8.037977200459484, −7.380642530932746, −7.018135135785367, −6.098387404860162, −5.899784558260253, −5.328916462342299, −4.751224005243037, −4.122483534317142, −3.840169744497886, −3.108107274505609, −2.366639537558417, −1.689119859524538, −1.313100776738069, −0.6091156942774175,
0.6091156942774175, 1.313100776738069, 1.689119859524538, 2.366639537558417, 3.108107274505609, 3.840169744497886, 4.122483534317142, 4.751224005243037, 5.328916462342299, 5.899784558260253, 6.098387404860162, 7.018135135785367, 7.380642530932746, 8.037977200459484, 8.320974465989526, 8.675523425425961, 9.308064489001227, 9.923676929672644, 10.36015577749858, 10.90734701469700, 11.19273387294886, 11.82158383289109, 12.02969207334776, 12.61767903361409, 13.17923286524785
