| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s − 3·7-s + 8-s + 9-s + 10-s + 11-s + 12-s − 3·14-s + 15-s + 16-s + 2·17-s + 18-s − 2·19-s + 20-s − 3·21-s + 22-s + 3·23-s + 24-s + 25-s + 27-s − 3·28-s + 29-s + 30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 1.13·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.301·11-s + 0.288·12-s − 0.801·14-s + 0.258·15-s + 1/4·16-s + 0.485·17-s + 0.235·18-s − 0.458·19-s + 0.223·20-s − 0.654·21-s + 0.213·22-s + 0.625·23-s + 0.204·24-s + 1/5·25-s + 0.192·27-s − 0.566·28-s + 0.185·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 - T \) | |
| 11 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 - T + p T^{2} \) | 1.29.ab |
| 31 | \( 1 + 7 T + p T^{2} \) | 1.31.h |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 - 11 T + p T^{2} \) | 1.43.al |
| 47 | \( 1 + 11 T + p T^{2} \) | 1.47.l |
| 53 | \( 1 - 11 T + p T^{2} \) | 1.53.al |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−14.53614314628493, −14.20187591650269, −13.52888063884677, −13.14780837598524, −12.79862906809239, −12.33315444089458, −11.70983278559033, −11.11695944853134, −10.43131240625813, −10.07601225886175, −9.522426981530344, −8.909590637770564, −8.635496457768491, −7.613577411331547, −7.276397289917524, −6.695042398331414, −6.063848665012141, −5.752004378905931, −4.946937837186412, −4.301985511752405, −3.709851891431660, −3.082746897729389, −2.732254231274159, −1.866310912672305, −1.199615423258650, 0,
1.199615423258650, 1.866310912672305, 2.732254231274159, 3.082746897729389, 3.709851891431660, 4.301985511752405, 4.946937837186412, 5.752004378905931, 6.063848665012141, 6.695042398331414, 7.276397289917524, 7.613577411331547, 8.635496457768491, 8.909590637770564, 9.522426981530344, 10.07601225886175, 10.43131240625813, 11.11695944853134, 11.70983278559033, 12.33315444089458, 12.79862906809239, 13.14780837598524, 13.52888063884677, 14.20187591650269, 14.53614314628493
