| L(s) = 1 | − 4·5-s + 4·7-s − 4·11-s − 8·23-s + 11·25-s − 8·29-s + 4·31-s − 16·35-s + 6·37-s + 12·41-s − 8·43-s + 4·47-s + 9·49-s + 16·55-s + 4·59-s + 2·61-s + 8·67-s + 4·71-s + 10·73-s − 16·77-s + 4·79-s + 12·83-s − 12·89-s − 14·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 1.51·7-s − 1.20·11-s − 1.66·23-s + 11/5·25-s − 1.48·29-s + 0.718·31-s − 2.70·35-s + 0.986·37-s + 1.87·41-s − 1.21·43-s + 0.583·47-s + 9/7·49-s + 2.15·55-s + 0.520·59-s + 0.256·61-s + 0.977·67-s + 0.474·71-s + 1.17·73-s − 1.82·77-s + 0.450·79-s + 1.31·83-s − 1.27·89-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 97344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.199528962\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.199528962\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 4 T + p T^{2} \) | 1.5.e |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 12 T + p T^{2} \) | 1.41.am |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 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
−13.83254277143808, −13.30599638150495, −12.57041591565092, −12.33798013994963, −11.69448597578969, −11.31905158714710, −11.01380628400506, −10.58074385202460, −9.862708462772563, −9.293455092617297, −8.400302760513337, −8.169315469693571, −7.879119903311633, −7.487583893919116, −6.941265614862486, −6.058773193570755, −5.454703457363775, −4.933961111888892, −4.398064048446399, −3.973176807716695, −3.461942454287084, −2.506070464316867, −2.125892233533521, −1.111570898352503, −0.3853883864814317,
0.3853883864814317, 1.111570898352503, 2.125892233533521, 2.506070464316867, 3.461942454287084, 3.973176807716695, 4.398064048446399, 4.933961111888892, 5.454703457363775, 6.058773193570755, 6.941265614862486, 7.487583893919116, 7.879119903311633, 8.169315469693571, 8.400302760513337, 9.293455092617297, 9.862708462772563, 10.58074385202460, 11.01380628400506, 11.31905158714710, 11.69448597578969, 12.33798013994963, 12.57041591565092, 13.30599638150495, 13.83254277143808
