| L(s) = 1 | − 5-s − 4·11-s + 2·17-s − 4·19-s − 8·23-s + 25-s + 6·29-s + 10·37-s + 6·41-s − 4·43-s − 7·49-s − 6·53-s + 4·55-s + 12·59-s − 10·61-s + 12·67-s − 12·71-s + 10·73-s + 8·79-s + 16·83-s − 2·85-s + 14·89-s + 4·95-s + 2·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.20·11-s + 0.485·17-s − 0.917·19-s − 1.66·23-s + 1/5·25-s + 1.11·29-s + 1.64·37-s + 0.937·41-s − 0.609·43-s − 49-s − 0.824·53-s + 0.539·55-s + 1.56·59-s − 1.28·61-s + 1.46·67-s − 1.42·71-s + 1.17·73-s + 0.900·79-s + 1.75·83-s − 0.216·85-s + 1.48·89-s + 0.410·95-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 243360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.375332442\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.375332442\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 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
−12.93195873237334, −12.35052842498801, −12.05826551427896, −11.49913076783771, −10.94485584192621, −10.63452679554344, −10.07648717797815, −9.735406212476130, −9.208608844059391, −8.429565431279994, −8.127581848896697, −7.819455413030871, −7.401462240164188, −6.465225488022731, −6.359211802643724, −5.719305717718048, −5.095219941604945, −4.641358181729374, −4.157759429422522, −3.566170805075589, −2.986894353539280, −2.356230974406878, −1.984903622665048, −0.9927852369841457, −0.3636002755899195,
0.3636002755899195, 0.9927852369841457, 1.984903622665048, 2.356230974406878, 2.986894353539280, 3.566170805075589, 4.157759429422522, 4.641358181729374, 5.095219941604945, 5.719305717718048, 6.359211802643724, 6.465225488022731, 7.401462240164188, 7.819455413030871, 8.127581848896697, 8.429565431279994, 9.208608844059391, 9.735406212476130, 10.07648717797815, 10.63452679554344, 10.94485584192621, 11.49913076783771, 12.05826551427896, 12.35052842498801, 12.93195873237334
