| L(s) = 1 | − 3-s + 5-s − 4·7-s + 9-s − 11-s − 15-s − 2·17-s + 2·19-s + 4·21-s + 4·23-s + 25-s − 27-s + 4·31-s + 33-s − 4·35-s + 2·37-s + 12·41-s − 4·43-s + 45-s − 8·47-s + 9·49-s + 2·51-s + 10·53-s − 55-s − 2·57-s − 12·59-s − 2·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 1.51·7-s + 1/3·9-s − 0.301·11-s − 0.258·15-s − 0.485·17-s + 0.458·19-s + 0.872·21-s + 0.834·23-s + 1/5·25-s − 0.192·27-s + 0.718·31-s + 0.174·33-s − 0.676·35-s + 0.328·37-s + 1.87·41-s − 0.609·43-s + 0.149·45-s − 1.16·47-s + 9/7·49-s + 0.280·51-s + 1.37·53-s − 0.134·55-s − 0.264·57-s − 1.56·59-s − 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 + T \) | |
| 5 | \( 1 - T \) | |
| 11 | \( 1 + T \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 12 T + p T^{2} \) | 1.41.am |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + p T^{2} \) | 1.73.a |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−7.66162202809526546629380408828, −6.95035449385736333746409121138, −6.34245820867533170342910592729, −5.83108701901217886773344156342, −5.00419752413212668336861734416, −4.16983221612528042124919434845, −3.15841946538787660844264127986, −2.54007590143305254280240974299, −1.16944168080982520600245832446, 0,
1.16944168080982520600245832446, 2.54007590143305254280240974299, 3.15841946538787660844264127986, 4.16983221612528042124919434845, 5.00419752413212668336861734416, 5.83108701901217886773344156342, 6.34245820867533170342910592729, 6.95035449385736333746409121138, 7.66162202809526546629380408828
