| L(s) = 1 | + 3·5-s + 2·11-s − 13-s − 7·19-s − 7·23-s + 4·25-s + 3·29-s − 5·31-s − 4·37-s − 6·41-s − 11·43-s + 3·47-s − 9·53-s + 6·55-s − 8·59-s − 14·61-s − 3·65-s + 2·67-s − 3·73-s + 3·79-s + 9·83-s − 9·89-s − 21·95-s − 7·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 1.34·5-s + 0.603·11-s − 0.277·13-s − 1.60·19-s − 1.45·23-s + 4/5·25-s + 0.557·29-s − 0.898·31-s − 0.657·37-s − 0.937·41-s − 1.67·43-s + 0.437·47-s − 1.23·53-s + 0.809·55-s − 1.04·59-s − 1.79·61-s − 0.372·65-s + 0.244·67-s − 0.351·73-s + 0.337·79-s + 0.987·83-s − 0.953·89-s − 2.15·95-s − 0.710·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 366912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 366912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8617402119\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8617402119\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 + 7 T + p T^{2} \) | 1.23.h |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 11 T + p T^{2} \) | 1.43.l |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 3 T + p T^{2} \) | 1.73.d |
| 79 | \( 1 - 3 T + p T^{2} \) | 1.79.ad |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 89 | \( 1 + 9 T + p T^{2} \) | 1.89.j |
| 97 | \( 1 + 7 T + p T^{2} \) | 1.97.h |
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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
−12.42441154482336, −12.12437604737511, −11.77983809565209, −10.83302918474304, −10.78101058391707, −10.23305856279418, −9.726343816684615, −9.447017318065438, −8.955120547226631, −8.375999236519210, −8.116080990656264, −7.406983641262201, −6.742762814958823, −6.431933095357394, −6.139463474629305, −5.521473823550708, −5.111073132569119, −4.445349221490496, −4.091353324248487, −3.358547989280884, −2.856195564608172, −2.078698933463585, −1.777675316653977, −1.419790914169920, −0.2143316395005697,
0.2143316395005697, 1.419790914169920, 1.777675316653977, 2.078698933463585, 2.856195564608172, 3.358547989280884, 4.091353324248487, 4.445349221490496, 5.111073132569119, 5.521473823550708, 6.139463474629305, 6.431933095357394, 6.742762814958823, 7.406983641262201, 8.116080990656264, 8.375999236519210, 8.955120547226631, 9.447017318065438, 9.726343816684615, 10.23305856279418, 10.78101058391707, 10.83302918474304, 11.77983809565209, 12.12437604737511, 12.42441154482336
