| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 7-s + 8-s + 9-s + 10-s − 2·11-s + 12-s + 14-s + 15-s + 16-s − 3·17-s + 18-s + 6·19-s + 20-s + 21-s − 2·22-s − 4·23-s + 24-s + 25-s + 27-s + 28-s − 5·29-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 + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.603·11-s + 0.288·12-s + 0.267·14-s + 0.258·15-s + 1/4·16-s − 0.727·17-s + 0.235·18-s + 1.37·19-s + 0.223·20-s + 0.218·21-s − 0.426·22-s − 0.834·23-s + 0.204·24-s + 1/5·25-s + 0.192·27-s + 0.188·28-s − 0.928·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63870 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63870 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 2129 | \( 1 - T \) | |
| good | 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 + 7 T + p T^{2} \) | 1.41.h |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 3 T + p T^{2} \) | 1.79.ad |
| 83 | \( 1 + 7 T + p T^{2} \) | 1.83.h |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−14.33930761368341, −13.88057900644151, −13.64720248120579, −13.10969904703992, −12.63011805227235, −12.03622691746783, −11.48672329078397, −11.10124412906791, −10.33933161643355, −10.01435447978790, −9.407303926425262, −8.850343502579492, −8.246850380192400, −7.651035289868516, −7.326709593584414, −6.623526567050013, −5.944709454221132, −5.543223641230460, −4.851405385006986, −4.401355308910228, −3.729889725386094, −2.969607384329580, −2.620224609563513, −1.789789860875229, −1.306472094661612, 0,
1.306472094661612, 1.789789860875229, 2.620224609563513, 2.969607384329580, 3.729889725386094, 4.401355308910228, 4.851405385006986, 5.543223641230460, 5.944709454221132, 6.623526567050013, 7.326709593584414, 7.651035289868516, 8.246850380192400, 8.850343502579492, 9.407303926425262, 10.01435447978790, 10.33933161643355, 11.10124412906791, 11.48672329078397, 12.03622691746783, 12.63011805227235, 13.10969904703992, 13.64720248120579, 13.88057900644151, 14.33930761368341
