| L(s) = 1 | − 5-s + 7-s + 2·11-s + 3·17-s + 6·19-s − 4·23-s − 4·25-s − 2·29-s + 4·31-s − 35-s − 3·37-s + 5·43-s − 13·47-s − 6·49-s − 12·53-s − 2·55-s + 10·59-s − 8·61-s − 2·67-s + 5·71-s + 10·73-s + 2·77-s + 4·79-s − 3·85-s + 6·89-s − 6·95-s − 14·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.377·7-s + 0.603·11-s + 0.727·17-s + 1.37·19-s − 0.834·23-s − 4/5·25-s − 0.371·29-s + 0.718·31-s − 0.169·35-s − 0.493·37-s + 0.762·43-s − 1.89·47-s − 6/7·49-s − 1.64·53-s − 0.269·55-s + 1.30·59-s − 1.02·61-s − 0.244·67-s + 0.593·71-s + 1.17·73-s + 0.227·77-s + 0.450·79-s − 0.325·85-s + 0.635·89-s − 0.615·95-s − 1.42·97-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 3 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 5 T + p T^{2} \) | 1.43.af |
| 47 | \( 1 + 13 T + p T^{2} \) | 1.47.n |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 5 T + p T^{2} \) | 1.71.af |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 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
−15.83430013531315, −15.07896759828337, −14.58367397579707, −14.03747614627018, −13.72473348246403, −12.95271923182586, −12.22768714795330, −11.93937470777804, −11.37988991044418, −10.96184397755244, −10.04143200537696, −9.676829884690844, −9.195918565509043, −8.237851572674227, −7.935910214293559, −7.466931689473397, −6.621172208536404, −6.137165650579257, −5.307522612086091, −4.879794765124937, −3.967301598190782, −3.548462185251149, −2.797079105093174, −1.762617081161346, −1.151362713852142, 0,
1.151362713852142, 1.762617081161346, 2.797079105093174, 3.548462185251149, 3.967301598190782, 4.879794765124937, 5.307522612086091, 6.137165650579257, 6.621172208536404, 7.466931689473397, 7.935910214293559, 8.237851572674227, 9.195918565509043, 9.676829884690844, 10.04143200537696, 10.96184397755244, 11.37988991044418, 11.93937470777804, 12.22768714795330, 12.95271923182586, 13.72473348246403, 14.03747614627018, 14.58367397579707, 15.07896759828337, 15.83430013531315
