| L(s) = 1 | + 4·5-s + 2·11-s − 2·17-s + 8·19-s + 4·23-s + 11·25-s + 6·29-s − 4·31-s − 6·37-s − 12·41-s − 4·43-s + 6·47-s − 7·49-s + 2·53-s + 8·55-s + 14·59-s + 10·61-s − 4·67-s − 2·71-s + 2·73-s + 8·79-s − 14·83-s − 8·85-s + 32·95-s + 10·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 0.603·11-s − 0.485·17-s + 1.83·19-s + 0.834·23-s + 11/5·25-s + 1.11·29-s − 0.718·31-s − 0.986·37-s − 1.87·41-s − 0.609·43-s + 0.875·47-s − 49-s + 0.274·53-s + 1.07·55-s + 1.82·59-s + 1.28·61-s − 0.488·67-s − 0.237·71-s + 0.234·73-s + 0.900·79-s − 1.53·83-s − 0.867·85-s + 3.28·95-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.155291504\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.155291504\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 14 T + p T^{2} \) | 1.59.ao |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−15.45366243484201, −14.55363674874730, −14.38293834162846, −13.65896095565994, −13.44991251332703, −12.88953910603900, −12.14684350920028, −11.63465203649663, −11.03901285696574, −10.22390095010235, −9.973815095076147, −9.451344605275619, −8.772442217733458, −8.523884284724621, −7.357279824374776, −6.845339530714320, −6.466759033716218, −5.605744036955208, −5.249163376435370, −4.721874514190358, −3.576062296204987, −3.041580830826212, −2.198789871128608, −1.553413150550353, −0.8629376023954402,
0.8629376023954402, 1.553413150550353, 2.198789871128608, 3.041580830826212, 3.576062296204987, 4.721874514190358, 5.249163376435370, 5.605744036955208, 6.466759033716218, 6.845339530714320, 7.357279824374776, 8.523884284724621, 8.772442217733458, 9.451344605275619, 9.973815095076147, 10.22390095010235, 11.03901285696574, 11.63465203649663, 12.14684350920028, 12.88953910603900, 13.44991251332703, 13.65896095565994, 14.38293834162846, 14.55363674874730, 15.45366243484201
