| L(s) = 1 | + 3·5-s + 2·7-s + 2·11-s + 2·17-s − 5·19-s + 8·23-s + 4·25-s + 8·29-s + 5·31-s + 6·35-s + 2·37-s − 6·41-s − 4·43-s + 6·47-s − 3·49-s + 53-s + 6·55-s − 3·59-s − 13·67-s − 8·71-s + 9·73-s + 4·77-s + 79-s + 2·83-s + 6·85-s + 9·89-s − 15·95-s + ⋯ |
| L(s) = 1 | + 1.34·5-s + 0.755·7-s + 0.603·11-s + 0.485·17-s − 1.14·19-s + 1.66·23-s + 4/5·25-s + 1.48·29-s + 0.898·31-s + 1.01·35-s + 0.328·37-s − 0.937·41-s − 0.609·43-s + 0.875·47-s − 3/7·49-s + 0.137·53-s + 0.809·55-s − 0.390·59-s − 1.58·67-s − 0.949·71-s + 1.05·73-s + 0.455·77-s + 0.112·79-s + 0.219·83-s + 0.650·85-s + 0.953·89-s − 1.53·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 228672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.461704714\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.461704714\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 397 | \( 1 + T \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - T + p T^{2} \) | 1.53.ab |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 13 T + p T^{2} \) | 1.67.n |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 9 T + p T^{2} \) | 1.73.aj |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 - 2 T + p T^{2} \) | 1.83.ac |
| 89 | \( 1 - 9 T + p T^{2} \) | 1.89.aj |
| 97 | \( 1 - 9 T + p T^{2} \) | 1.97.aj |
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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
−13.02421159444503, −12.54908820309364, −11.89654326088479, −11.66172807366701, −10.99924038140873, −10.42747982058781, −10.29277486541964, −9.711882280150105, −9.095141538573224, −8.782323013936080, −8.391110843571486, −7.737049000212256, −7.192860801058915, −6.488194652407926, −6.355113367105632, −5.805746460350990, −5.023130118954418, −4.854793206614482, −4.319102097524503, −3.482053096462610, −2.904884406404617, −2.372053336359378, −1.733481954824676, −1.277540074310368, −0.6699680513973910,
0.6699680513973910, 1.277540074310368, 1.733481954824676, 2.372053336359378, 2.904884406404617, 3.482053096462610, 4.319102097524503, 4.854793206614482, 5.023130118954418, 5.805746460350990, 6.355113367105632, 6.488194652407926, 7.192860801058915, 7.737049000212256, 8.391110843571486, 8.782323013936080, 9.095141538573224, 9.711882280150105, 10.29277486541964, 10.42747982058781, 10.99924038140873, 11.66172807366701, 11.89654326088479, 12.54908820309364, 13.02421159444503
