| L(s) = 1 | + 5-s − 7-s − 3·11-s − 4·13-s − 17-s + 5·19-s + 6·23-s + 25-s + 3·29-s − 4·31-s − 35-s + 5·37-s − 9·41-s − 10·43-s + 3·47-s − 6·49-s − 3·53-s − 3·55-s − 12·59-s − 10·61-s − 4·65-s − 10·67-s − 73-s + 3·77-s − 4·79-s + 12·83-s − 85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.377·7-s − 0.904·11-s − 1.10·13-s − 0.242·17-s + 1.14·19-s + 1.25·23-s + 1/5·25-s + 0.557·29-s − 0.718·31-s − 0.169·35-s + 0.821·37-s − 1.40·41-s − 1.52·43-s + 0.437·47-s − 6/7·49-s − 0.412·53-s − 0.404·55-s − 1.56·59-s − 1.28·61-s − 0.496·65-s − 1.22·67-s − 0.117·73-s + 0.341·77-s − 0.450·79-s + 1.31·83-s − 0.108·85-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 17 | \( 1 + T \) | |
| good | 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 5 T + p T^{2} \) | 1.37.af |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 + 3 T + p T^{2} \) | 1.53.d |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 10 T + p T^{2} \) | 1.67.k |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + T + p T^{2} \) | 1.73.b |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 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
−8.310200641352159456350389711985, −7.49687733509651346873609559165, −6.93021424771158483571966975624, −6.03689669237326195768388300241, −5.11550548700533921974608157798, −4.74062296251735093151905532428, −3.24964758538908192758533226265, −2.74398924221222077751749040496, −1.53664710484848044917006908777, 0,
1.53664710484848044917006908777, 2.74398924221222077751749040496, 3.24964758538908192758533226265, 4.74062296251735093151905532428, 5.11550548700533921974608157798, 6.03689669237326195768388300241, 6.93021424771158483571966975624, 7.49687733509651346873609559165, 8.310200641352159456350389711985
