| L(s) = 1 | − 5-s + 4·7-s − 4·11-s + 6·13-s − 17-s + 4·19-s − 4·23-s + 25-s + 6·29-s − 2·31-s − 4·35-s + 10·41-s − 4·43-s + 2·47-s + 9·49-s − 2·53-s + 4·55-s − 10·61-s − 6·65-s − 4·67-s + 16·71-s − 8·73-s − 16·77-s + 10·79-s + 10·83-s + 85-s + 24·91-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.51·7-s − 1.20·11-s + 1.66·13-s − 0.242·17-s + 0.917·19-s − 0.834·23-s + 1/5·25-s + 1.11·29-s − 0.359·31-s − 0.676·35-s + 1.56·41-s − 0.609·43-s + 0.291·47-s + 9/7·49-s − 0.274·53-s + 0.539·55-s − 1.28·61-s − 0.744·65-s − 0.488·67-s + 1.89·71-s − 0.936·73-s − 1.82·77-s + 1.12·79-s + 1.09·83-s + 0.108·85-s + 2.51·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3060 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3060 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.099713479\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.099713479\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 17 | \( 1 + T \) | |
| good | 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + p T^{2} \) | 1.37.a |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 - 10 T + p T^{2} \) | 1.83.ak |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 16 T + p T^{2} \) | 1.97.aq |
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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
−8.475225904347930953202080845574, −7.970268829306120912317948428178, −7.54460349786155851189971750256, −6.37731656427454218605614709307, −5.56367536685584858158354287801, −4.84226702337758616223632806375, −4.08886985666447118980096603047, −3.10645511772208776776866655724, −1.97794887158850689743669643855, −0.926045851906879699244746082283,
0.926045851906879699244746082283, 1.97794887158850689743669643855, 3.10645511772208776776866655724, 4.08886985666447118980096603047, 4.84226702337758616223632806375, 5.56367536685584858158354287801, 6.37731656427454218605614709307, 7.54460349786155851189971750256, 7.970268829306120912317948428178, 8.475225904347930953202080845574
