| L(s) = 1 | + 2·3-s + 5-s + 9-s + 11-s + 4·13-s + 2·15-s − 4·17-s + 4·19-s + 2·23-s + 25-s − 4·27-s + 6·29-s + 4·31-s + 2·33-s + 2·37-s + 8·39-s − 2·41-s + 4·43-s + 45-s − 2·47-s − 7·49-s − 8·51-s + 2·53-s + 55-s + 8·57-s − 4·59-s − 2·61-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.447·5-s + 1/3·9-s + 0.301·11-s + 1.10·13-s + 0.516·15-s − 0.970·17-s + 0.917·19-s + 0.417·23-s + 1/5·25-s − 0.769·27-s + 1.11·29-s + 0.718·31-s + 0.348·33-s + 0.328·37-s + 1.28·39-s − 0.312·41-s + 0.609·43-s + 0.149·45-s − 0.291·47-s − 49-s − 1.12·51-s + 0.274·53-s + 0.134·55-s + 1.05·57-s − 0.520·59-s − 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.970482138\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.970482138\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 11 | \( 1 - T \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−9.277658051155937282721557993057, −8.475290682915387059417549067236, −8.044584521886118214744806811584, −6.88816956948104955344182232976, −6.24489673097193351654764436856, −5.18423643032243955059797257659, −4.12137302579351488766401281913, −3.22331889815417909127040494114, −2.42572981750989013699213972497, −1.24575535965830080961625131586,
1.24575535965830080961625131586, 2.42572981750989013699213972497, 3.22331889815417909127040494114, 4.12137302579351488766401281913, 5.18423643032243955059797257659, 6.24489673097193351654764436856, 6.88816956948104955344182232976, 8.044584521886118214744806811584, 8.475290682915387059417549067236, 9.277658051155937282721557993057
