| L(s) = 1 | − 5-s − 4·7-s + 2·11-s − 4·13-s − 17-s + 5·19-s − 5·23-s + 25-s + 8·29-s + 7·31-s + 4·35-s + 6·37-s − 6·41-s + 2·43-s − 8·47-s + 9·49-s + 9·53-s − 2·55-s + 4·59-s − 13·61-s + 4·65-s + 10·67-s + 6·71-s − 6·73-s − 8·77-s + 9·79-s − 17·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.51·7-s + 0.603·11-s − 1.10·13-s − 0.242·17-s + 1.14·19-s − 1.04·23-s + 1/5·25-s + 1.48·29-s + 1.25·31-s + 0.676·35-s + 0.986·37-s − 0.937·41-s + 0.304·43-s − 1.16·47-s + 9/7·49-s + 1.23·53-s − 0.269·55-s + 0.520·59-s − 1.66·61-s + 0.496·65-s + 1.22·67-s + 0.712·71-s − 0.702·73-s − 0.911·77-s + 1.01·79-s − 1.86·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + T + p T^{2} \) | 1.17.b |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 + 5 T + p T^{2} \) | 1.23.f |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 - 7 T + p T^{2} \) | 1.31.ah |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 13 T + p T^{2} \) | 1.61.n |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 9 T + p T^{2} \) | 1.79.aj |
| 83 | \( 1 + 17 T + p T^{2} \) | 1.83.r |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−7.39876087643947307363818415975, −6.56456120519014125283924210247, −6.37581007247427526629920238830, −5.33132767734034622392466185980, −4.55156179433330296239918727357, −3.81751554759938339802856258326, −3.04764875897694377170889535956, −2.46788403321754892094543922736, −1.04061506504350801766091790120, 0,
1.04061506504350801766091790120, 2.46788403321754892094543922736, 3.04764875897694377170889535956, 3.81751554759938339802856258326, 4.55156179433330296239918727357, 5.33132767734034622392466185980, 6.37581007247427526629920238830, 6.56456120519014125283924210247, 7.39876087643947307363818415975
