| L(s) = 1 | + 2-s + 3-s + 4-s + 4·5-s + 6-s − 4·7-s + 8-s + 9-s + 4·10-s + 11-s + 12-s − 13-s − 4·14-s + 4·15-s + 16-s + 4·17-s + 18-s − 4·19-s + 4·20-s − 4·21-s + 22-s + 8·23-s + 24-s + 11·25-s − 26-s + 27-s − 4·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.78·5-s + 0.408·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s + 1.26·10-s + 0.301·11-s + 0.288·12-s − 0.277·13-s − 1.06·14-s + 1.03·15-s + 1/4·16-s + 0.970·17-s + 0.235·18-s − 0.917·19-s + 0.894·20-s − 0.872·21-s + 0.213·22-s + 1.66·23-s + 0.204·24-s + 11/5·25-s − 0.196·26-s + 0.192·27-s − 0.755·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 858 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 858 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.418346408\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.418346408\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 3 | \( 1 - T \) | |
| 11 | \( 1 - T \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 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
−10.20217929164752063773669448388, −9.308592334120201396838900493464, −8.887860262586512140854508642144, −7.22619514679261276402292453329, −6.58410892357960633880686900153, −5.84003805364552851081046917817, −4.98774365097387991463126539862, −3.48962054325808624486728186620, −2.79700721288937028347473279893, −1.65926995200329057960656430027,
1.65926995200329057960656430027, 2.79700721288937028347473279893, 3.48962054325808624486728186620, 4.98774365097387991463126539862, 5.84003805364552851081046917817, 6.58410892357960633880686900153, 7.22619514679261276402292453329, 8.887860262586512140854508642144, 9.308592334120201396838900493464, 10.20217929164752063773669448388
