| L(s) = 1 | + 2·7-s − 17-s − 4·19-s + 6·29-s + 4·31-s + 10·37-s − 12·41-s − 10·43-s + 8·47-s − 3·49-s + 6·53-s − 6·59-s − 6·61-s + 6·67-s + 2·71-s − 12·73-s + 8·79-s + 2·89-s + 16·97-s + 101-s + 103-s + 107-s + 109-s + 113-s − 2·119-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 0.242·17-s − 0.917·19-s + 1.11·29-s + 0.718·31-s + 1.64·37-s − 1.87·41-s − 1.52·43-s + 1.16·47-s − 3/7·49-s + 0.824·53-s − 0.781·59-s − 0.768·61-s + 0.733·67-s + 0.237·71-s − 1.40·73-s + 0.900·79-s + 0.211·89-s + 1.62·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s − 0.183·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 244800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 244800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.692033566\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.692033566\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 - 6 T + p T^{2} \) | 1.67.ag |
| 71 | \( 1 - 2 T + p T^{2} \) | 1.71.ac |
| 73 | \( 1 + 12 T + p T^{2} \) | 1.73.m |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 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
−12.94937638466764, −12.30450973266947, −11.85990719536913, −11.62421841184378, −10.94182379622376, −10.66072973774595, −9.985337612584433, −9.853898120454804, −8.951063210215732, −8.635086788348639, −8.241073679303650, −7.779533346080726, −7.213705254544650, −6.632454976935612, −6.253019585338451, −5.753902738994869, −4.994150300631479, −4.616902249171668, −4.343256337704977, −3.472388839941379, −3.052168972494544, −2.254559904556252, −1.899183994722054, −1.129983977546049, −0.4774065885559407,
0.4774065885559407, 1.129983977546049, 1.899183994722054, 2.254559904556252, 3.052168972494544, 3.472388839941379, 4.343256337704977, 4.616902249171668, 4.994150300631479, 5.753902738994869, 6.253019585338451, 6.632454976935612, 7.213705254544650, 7.779533346080726, 8.241073679303650, 8.635086788348639, 8.951063210215732, 9.853898120454804, 9.985337612584433, 10.66072973774595, 10.94182379622376, 11.62421841184378, 11.85990719536913, 12.30450973266947, 12.94937638466764
