| L(s) = 1 | + 2-s + 4-s − 5-s − 7-s + 8-s − 10-s − 13-s − 14-s + 16-s − 3·17-s + 5·19-s − 20-s − 23-s + 25-s − 26-s − 28-s + 6·29-s − 10·31-s + 32-s − 3·34-s + 35-s − 10·37-s + 5·38-s − 40-s + 6·41-s − 43-s − 46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.377·7-s + 0.353·8-s − 0.316·10-s − 0.277·13-s − 0.267·14-s + 1/4·16-s − 0.727·17-s + 1.14·19-s − 0.223·20-s − 0.208·23-s + 1/5·25-s − 0.196·26-s − 0.188·28-s + 1.11·29-s − 1.79·31-s + 0.176·32-s − 0.514·34-s + 0.169·35-s − 1.64·37-s + 0.811·38-s − 0.158·40-s + 0.937·41-s − 0.152·43-s − 0.147·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 23 | \( 1 + T \) | |
| good | 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 + 9 T + p T^{2} \) | 1.47.j |
| 53 | \( 1 + 3 T + p T^{2} \) | 1.53.d |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 9 T + p T^{2} \) | 1.71.aj |
| 73 | \( 1 + 7 T + p T^{2} \) | 1.73.h |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
| 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
−7.52439263365442147750318270815, −6.92620379729250966464225740896, −6.33533290579671143567009426794, −5.36523146894131907230426334646, −4.90193298235546671738135983492, −3.93821504103483897115872873107, −3.35967563259678248449739699961, −2.52970737779096206015496269051, −1.45981677192941262208840341188, 0,
1.45981677192941262208840341188, 2.52970737779096206015496269051, 3.35967563259678248449739699961, 3.93821504103483897115872873107, 4.90193298235546671738135983492, 5.36523146894131907230426334646, 6.33533290579671143567009426794, 6.92620379729250966464225740896, 7.52439263365442147750318270815
