| L(s) = 1 | + 2-s + 4-s − 2·5-s + 8-s − 2·10-s − 3·11-s + 13-s + 16-s + 17-s + 3·19-s − 2·20-s − 3·22-s − 25-s + 26-s − 3·29-s + 4·31-s + 32-s + 34-s − 2·37-s + 3·38-s − 2·40-s + 2·41-s + 6·43-s − 3·44-s − 9·47-s − 50-s + 52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.894·5-s + 0.353·8-s − 0.632·10-s − 0.904·11-s + 0.277·13-s + 1/4·16-s + 0.242·17-s + 0.688·19-s − 0.447·20-s − 0.639·22-s − 1/5·25-s + 0.196·26-s − 0.557·29-s + 0.718·31-s + 0.176·32-s + 0.171·34-s − 0.328·37-s + 0.486·38-s − 0.316·40-s + 0.312·41-s + 0.914·43-s − 0.452·44-s − 1.31·47-s − 0.141·50-s + 0.138·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11466 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11466 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 17 | \( 1 - T + p T^{2} \) | 1.17.ab |
| 19 | \( 1 - 3 T + p T^{2} \) | 1.19.ad |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + 9 T + p T^{2} \) | 1.47.j |
| 53 | \( 1 + T + p T^{2} \) | 1.53.b |
| 59 | \( 1 - 11 T + p T^{2} \) | 1.59.al |
| 61 | \( 1 + 11 T + p T^{2} \) | 1.61.l |
| 67 | \( 1 + 7 T + p T^{2} \) | 1.67.h |
| 71 | \( 1 + 15 T + p T^{2} \) | 1.71.p |
| 73 | \( 1 - 12 T + p T^{2} \) | 1.73.am |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.28280658372378, −16.16562658360320, −15.56974316136697, −15.03766566601460, −14.54617427462424, −13.74900096307460, −13.37758563513532, −12.73672057635469, −12.10465099820924, −11.68403054429668, −11.07356777735378, −10.52260667309293, −9.854700666152639, −9.102472828726552, −8.241851171629212, −7.750945339585402, −7.323326098262351, −6.470496331845077, −5.763890205489175, −5.128588346733742, −4.463252225824166, −3.732162667484577, −3.134763030082304, −2.357402150859162, −1.237410520427512, 0,
1.237410520427512, 2.357402150859162, 3.134763030082304, 3.732162667484577, 4.463252225824166, 5.128588346733742, 5.763890205489175, 6.470496331845077, 7.323326098262351, 7.750945339585402, 8.241851171629212, 9.102472828726552, 9.854700666152639, 10.52260667309293, 11.07356777735378, 11.68403054429668, 12.10465099820924, 12.73672057635469, 13.37758563513532, 13.74900096307460, 14.54617427462424, 15.03766566601460, 15.56974316136697, 16.16562658360320, 16.28280658372378
