| L(s) = 1 | − 4·7-s − 2·11-s + 13-s + 2·17-s − 6·19-s − 6·23-s − 2·29-s + 6·31-s + 2·37-s − 10·41-s − 10·43-s + 12·47-s + 9·49-s + 2·53-s + 10·59-s + 2·61-s − 12·67-s + 10·71-s − 10·73-s + 8·77-s + 4·79-s + 14·89-s − 4·91-s − 14·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 0.603·11-s + 0.277·13-s + 0.485·17-s − 1.37·19-s − 1.25·23-s − 0.371·29-s + 1.07·31-s + 0.328·37-s − 1.56·41-s − 1.52·43-s + 1.75·47-s + 9/7·49-s + 0.274·53-s + 1.30·59-s + 0.256·61-s − 1.46·67-s + 1.18·71-s − 1.17·73-s + 0.911·77-s + 0.450·79-s + 1.48·89-s − 0.419·91-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4578111355\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4578111355\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−14.81468137510506, −13.84422532832874, −13.49911504187212, −13.17111486594036, −12.53501841981274, −12.09292080707274, −11.67983265462451, −10.77694230849397, −10.29262574499205, −10.05957953497791, −9.456081427576616, −8.801406298623342, −8.256963293522670, −7.832792286795169, −6.886417131769271, −6.653867720155729, −5.997655709213787, −5.557894498911867, −4.777021264043756, −3.924330981962506, −3.660601912669189, −2.760445854345499, −2.348615953540351, −1.362341114539833, −0.2385312691037813,
0.2385312691037813, 1.362341114539833, 2.348615953540351, 2.760445854345499, 3.660601912669189, 3.924330981962506, 4.777021264043756, 5.557894498911867, 5.997655709213787, 6.653867720155729, 6.886417131769271, 7.832792286795169, 8.256963293522670, 8.801406298623342, 9.456081427576616, 10.05957953497791, 10.29262574499205, 10.77694230849397, 11.67983265462451, 12.09292080707274, 12.53501841981274, 13.17111486594036, 13.49911504187212, 13.84422532832874, 14.81468137510506
