| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 7-s − 8-s − 2·9-s + 10-s − 12-s − 13-s + 14-s + 15-s + 16-s + 5·17-s + 2·18-s + 3·19-s − 20-s + 21-s − 4·23-s + 24-s + 25-s + 26-s + 5·27-s − 28-s − 29-s − 30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s − 2/3·9-s + 0.316·10-s − 0.288·12-s − 0.277·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s + 1.21·17-s + 0.471·18-s + 0.688·19-s − 0.223·20-s + 0.218·21-s − 0.834·23-s + 0.204·24-s + 1/5·25-s + 0.196·26-s + 0.962·27-s − 0.188·28-s − 0.185·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5204867673\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5204867673\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 17 | \( 1 - 5 T + p T^{2} \) | 1.17.af |
| 19 | \( 1 - 3 T + p T^{2} \) | 1.19.ad |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 + 3 T + p T^{2} \) | 1.31.d |
| 37 | \( 1 + 9 T + p T^{2} \) | 1.37.j |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 5 T + p T^{2} \) | 1.53.af |
| 59 | \( 1 + 2 T + p T^{2} \) | 1.59.c |
| 61 | \( 1 - 13 T + p T^{2} \) | 1.61.an |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 7 T + p T^{2} \) | 1.71.ah |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 + 2 T + p T^{2} \) | 1.83.c |
| 89 | \( 1 + 9 T + p T^{2} \) | 1.89.j |
| 97 | \( 1 + p T^{2} \) | 1.97.a |
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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
−16.22718901791375, −15.60623114409000, −15.00611338204830, −14.31155800998380, −13.96543162271855, −13.07117377023394, −12.30306757726615, −12.01324460920014, −11.55960819366864, −10.85544516588223, −10.35250861747786, −9.754719174321908, −9.234739870456851, −8.399685209010341, −8.058050940662685, −7.258749825929194, −6.822962247191501, −5.966152859421078, −5.497733455013422, −4.882819347885195, −3.707172999912878, −3.281521397449646, −2.382781200583377, −1.354343969115436, −0.3784231093905544,
0.3784231093905544, 1.354343969115436, 2.382781200583377, 3.281521397449646, 3.707172999912878, 4.882819347885195, 5.497733455013422, 5.966152859421078, 6.822962247191501, 7.258749825929194, 8.058050940662685, 8.399685209010341, 9.234739870456851, 9.754719174321908, 10.35250861747786, 10.85544516588223, 11.55960819366864, 12.01324460920014, 12.30306757726615, 13.07117377023394, 13.96543162271855, 14.31155800998380, 15.00611338204830, 15.60623114409000, 16.22718901791375
