| L(s) = 1 | − 2-s + 4-s + 4·5-s − 8-s − 4·10-s + 5·11-s + 16-s − 5·17-s − 19-s + 4·20-s − 5·22-s − 6·23-s + 11·25-s − 8·31-s − 32-s + 5·34-s + 38-s − 4·40-s + 3·41-s − 9·43-s + 5·44-s + 6·46-s + 10·47-s − 7·49-s − 11·50-s − 6·53-s + 20·55-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.78·5-s − 0.353·8-s − 1.26·10-s + 1.50·11-s + 1/4·16-s − 1.21·17-s − 0.229·19-s + 0.894·20-s − 1.06·22-s − 1.25·23-s + 11/5·25-s − 1.43·31-s − 0.176·32-s + 0.857·34-s + 0.162·38-s − 0.632·40-s + 0.468·41-s − 1.37·43-s + 0.753·44-s + 0.884·46-s + 1.45·47-s − 49-s − 1.55·50-s − 0.824·53-s + 2.69·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136242 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136242 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 3 | \( 1 \) | |
| 29 | \( 1 \) | |
| good | 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + 5 T + p T^{2} \) | 1.17.f |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + p T^{2} \) | 1.37.a |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 + 9 T + p T^{2} \) | 1.43.j |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 7 T + p T^{2} \) | 1.59.ah |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 - 5 T + p T^{2} \) | 1.67.af |
| 71 | \( 1 - 2 T + p T^{2} \) | 1.71.ac |
| 73 | \( 1 - 13 T + p T^{2} \) | 1.73.an |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 + 15 T + p T^{2} \) | 1.83.p |
| 89 | \( 1 - 5 T + p T^{2} \) | 1.89.af |
| 97 | \( 1 + 9 T + p T^{2} \) | 1.97.j |
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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
−13.72854771503203, −13.25085533135825, −12.67333784885996, −12.33538672629828, −11.60438222405589, −11.14524549195072, −10.77752518442926, −10.12884047837501, −9.668980994429072, −9.429760774260595, −8.933079991509695, −8.530731136969129, −7.953795262612609, −7.045330282900894, −6.736698620530880, −6.350344526331248, −5.866436586373694, −5.349561876588330, −4.689480262934912, −3.927797623286103, −3.478976845640052, −2.408124098974263, −2.153464771794870, −1.616105051548209, −1.023670671541637, 0,
1.023670671541637, 1.616105051548209, 2.153464771794870, 2.408124098974263, 3.478976845640052, 3.927797623286103, 4.689480262934912, 5.349561876588330, 5.866436586373694, 6.350344526331248, 6.736698620530880, 7.045330282900894, 7.953795262612609, 8.530731136969129, 8.933079991509695, 9.429760774260595, 9.668980994429072, 10.12884047837501, 10.77752518442926, 11.14524549195072, 11.60438222405589, 12.33538672629828, 12.67333784885996, 13.25085533135825, 13.72854771503203
