| L(s) = 1 | − 3-s − 5-s + 2·7-s − 2·9-s + 13-s + 15-s − 4·17-s + 4·19-s − 2·21-s − 23-s + 25-s + 5·27-s − 3·29-s + 31-s − 2·35-s − 8·37-s − 39-s − 5·41-s + 6·43-s + 2·45-s − 9·47-s − 3·49-s + 4·51-s + 2·53-s − 4·57-s − 4·63-s − 65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.755·7-s − 2/3·9-s + 0.277·13-s + 0.258·15-s − 0.970·17-s + 0.917·19-s − 0.436·21-s − 0.208·23-s + 1/5·25-s + 0.962·27-s − 0.557·29-s + 0.179·31-s − 0.338·35-s − 1.31·37-s − 0.160·39-s − 0.780·41-s + 0.914·43-s + 0.298·45-s − 1.31·47-s − 3/7·49-s + 0.560·51-s + 0.274·53-s − 0.529·57-s − 0.503·63-s − 0.124·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 23 | \( 1 + T \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 - T + p T^{2} \) | 1.31.ab |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + 9 T + p T^{2} \) | 1.47.j |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 3 T + p T^{2} \) | 1.71.d |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−8.686483460882421074932983842715, −8.182391678331922016750663565187, −7.24891424844369717940081419261, −6.46355384215983737818519979347, −5.50009544222760448381601486169, −4.90400885608368494765313186973, −3.91591011669491162188697676251, −2.84409383180016604760332079864, −1.51878935676838052514160576069, 0,
1.51878935676838052514160576069, 2.84409383180016604760332079864, 3.91591011669491162188697676251, 4.90400885608368494765313186973, 5.50009544222760448381601486169, 6.46355384215983737818519979347, 7.24891424844369717940081419261, 8.182391678331922016750663565187, 8.686483460882421074932983842715
