| L(s) = 1 | − 3-s + 3·5-s + 9-s − 2·13-s − 3·15-s − 17-s − 4·19-s − 6·23-s + 4·25-s − 27-s − 8·31-s − 2·37-s + 2·39-s − 6·41-s + 9·43-s + 3·45-s + 9·47-s + 51-s + 8·53-s + 4·57-s + 5·59-s − 10·61-s − 6·65-s − 13·67-s + 6·69-s − 12·71-s + 12·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.34·5-s + 1/3·9-s − 0.554·13-s − 0.774·15-s − 0.242·17-s − 0.917·19-s − 1.25·23-s + 4/5·25-s − 0.192·27-s − 1.43·31-s − 0.328·37-s + 0.320·39-s − 0.937·41-s + 1.37·43-s + 0.447·45-s + 1.31·47-s + 0.140·51-s + 1.09·53-s + 0.529·57-s + 0.650·59-s − 1.28·61-s − 0.744·65-s − 1.58·67-s + 0.722·69-s − 1.42·71-s + 1.40·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 142296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 142296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.340674441\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.340674441\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + T + p T^{2} \) | 1.17.b |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 9 T + p T^{2} \) | 1.43.aj |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 53 | \( 1 - 8 T + p T^{2} \) | 1.53.ai |
| 59 | \( 1 - 5 T + p T^{2} \) | 1.59.af |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 13 T + p T^{2} \) | 1.67.n |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 12 T + p T^{2} \) | 1.73.am |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + T + p T^{2} \) | 1.83.b |
| 89 | \( 1 - 5 T + p T^{2} \) | 1.89.af |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−13.42785931701716, −12.94032034376467, −12.34204988048246, −12.14107502155917, −11.50057508830286, −10.76629762212296, −10.55321411592464, −10.11539741494162, −9.599294244924364, −9.059222855563358, −8.770281900602853, −7.980828365088244, −7.358330676658983, −6.999716539238877, −6.254369322750445, −5.976743570720915, −5.519393130806207, −5.038615333503136, −4.290369129707500, −3.946874191530243, −3.039430702055832, −2.237050885321610, −2.021652337865590, −1.322064385073380, −0.3436522943427194,
0.3436522943427194, 1.322064385073380, 2.021652337865590, 2.237050885321610, 3.039430702055832, 3.946874191530243, 4.290369129707500, 5.038615333503136, 5.519393130806207, 5.976743570720915, 6.254369322750445, 6.999716539238877, 7.358330676658983, 7.980828365088244, 8.770281900602853, 9.059222855563358, 9.599294244924364, 10.11539741494162, 10.55321411592464, 10.76629762212296, 11.50057508830286, 12.14107502155917, 12.34204988048246, 12.94032034376467, 13.42785931701716
