| L(s) = 1 | + 5·7-s − 6·11-s + 4·13-s + 7·17-s + 2·19-s − 2·29-s − 4·31-s − 10·37-s + 9·41-s − 3·47-s + 18·49-s + 10·53-s − 12·59-s + 2·67-s + 16·71-s + 5·73-s − 30·77-s + 15·79-s + 6·83-s + 3·89-s + 20·91-s − 13·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 1.88·7-s − 1.80·11-s + 1.10·13-s + 1.69·17-s + 0.458·19-s − 0.371·29-s − 0.718·31-s − 1.64·37-s + 1.40·41-s − 0.437·47-s + 18/7·49-s + 1.37·53-s − 1.56·59-s + 0.244·67-s + 1.89·71-s + 0.585·73-s − 3.41·77-s + 1.68·79-s + 0.658·83-s + 0.317·89-s + 2.09·91-s − 1.31·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 259200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 259200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.989515366\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.989515366\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| good | 7 | \( 1 - 5 T + p T^{2} \) | 1.7.af |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 7 T + p T^{2} \) | 1.17.ah |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 - 5 T + p T^{2} \) | 1.73.af |
| 79 | \( 1 - 15 T + p T^{2} \) | 1.79.ap |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 3 T + p T^{2} \) | 1.89.ad |
| 97 | \( 1 + 13 T + p T^{2} \) | 1.97.n |
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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
−12.72117080803078, −12.31010243526023, −11.96576877393281, −11.25526152626973, −10.99590862245509, −10.53725921556946, −10.34585451509748, −9.508435484947670, −9.096484247549193, −8.405175141924044, −8.036262521419443, −7.803710768575851, −7.427937262501032, −6.774640863433288, −5.901605179152543, −5.528361615822059, −5.121003418625966, −4.940899851247946, −3.974627013994539, −3.646950455743340, −2.947743918611363, −2.305684066882229, −1.748155601390224, −1.186975906264609, −0.5770027924641622,
0.5770027924641622, 1.186975906264609, 1.748155601390224, 2.305684066882229, 2.947743918611363, 3.646950455743340, 3.974627013994539, 4.940899851247946, 5.121003418625966, 5.528361615822059, 5.901605179152543, 6.774640863433288, 7.427937262501032, 7.803710768575851, 8.036262521419443, 8.405175141924044, 9.096484247549193, 9.508435484947670, 10.34585451509748, 10.53725921556946, 10.99590862245509, 11.25526152626973, 11.96576877393281, 12.31010243526023, 12.72117080803078
