| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s − 11-s − 12-s + 16-s + 17-s − 18-s + 8·19-s + 22-s + 4·23-s + 24-s − 27-s − 7·29-s + 2·31-s − 32-s + 33-s − 34-s + 36-s + 3·37-s − 8·38-s − 9·41-s − 43-s − 44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.301·11-s − 0.288·12-s + 1/4·16-s + 0.242·17-s − 0.235·18-s + 1.83·19-s + 0.213·22-s + 0.834·23-s + 0.204·24-s − 0.192·27-s − 1.29·29-s + 0.359·31-s − 0.176·32-s + 0.174·33-s − 0.171·34-s + 1/6·36-s + 0.493·37-s − 1.29·38-s − 1.40·41-s − 0.152·43-s − 0.150·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 3 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 - T + p T^{2} \) | 1.17.ab |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 7 T + p T^{2} \) | 1.29.h |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 - 11 T + p T^{2} \) | 1.47.al |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 3 T + p T^{2} \) | 1.71.ad |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 9 T + p T^{2} \) | 1.79.aj |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 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
−14.24389555093883, −13.54107444041086, −13.37421254419214, −12.57911058785141, −12.03224214361446, −11.81340272602589, −11.02245113310284, −10.92524545640458, −10.07785487145141, −9.873745966818560, −9.091406888078471, −8.950021523984097, −8.001895796310994, −7.571699836043136, −7.232814798429229, −6.639709304672446, −5.887434738678986, −5.551871058183517, −4.963377665128176, −4.344910446716267, −3.386434376801518, −3.098303509251759, −2.203660463509993, −1.424715910234113, −0.8762667233984382, 0,
0.8762667233984382, 1.424715910234113, 2.203660463509993, 3.098303509251759, 3.386434376801518, 4.344910446716267, 4.963377665128176, 5.551871058183517, 5.887434738678986, 6.639709304672446, 7.232814798429229, 7.571699836043136, 8.001895796310994, 8.950021523984097, 9.091406888078471, 9.873745966818560, 10.07785487145141, 10.92524545640458, 11.02245113310284, 11.81340272602589, 12.03224214361446, 12.57911058785141, 13.37421254419214, 13.54107444041086, 14.24389555093883
