| L(s) = 1 | + 3-s + 5-s + 2·7-s + 9-s − 11-s + 4·13-s + 15-s − 4·19-s + 2·21-s − 4·23-s + 25-s + 27-s + 2·29-s + 8·31-s − 33-s + 2·35-s + 2·37-s + 4·39-s + 6·41-s − 6·43-s + 45-s − 4·47-s − 3·49-s + 6·53-s − 55-s − 4·57-s − 4·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s − 0.301·11-s + 1.10·13-s + 0.258·15-s − 0.917·19-s + 0.436·21-s − 0.834·23-s + 1/5·25-s + 0.192·27-s + 0.371·29-s + 1.43·31-s − 0.174·33-s + 0.338·35-s + 0.328·37-s + 0.640·39-s + 0.937·41-s − 0.914·43-s + 0.149·45-s − 0.583·47-s − 3/7·49-s + 0.824·53-s − 0.134·55-s − 0.529·57-s − 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 381480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 381480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.212225044\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.212225044\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 - T \) | |
| 11 | \( 1 + T \) | |
| 17 | \( 1 \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−12.45447565786807, −12.09391055306701, −11.53328147007749, −10.97931320665268, −10.77469587467845, −10.14126961978609, −9.817173583118609, −9.295466095715192, −8.689254445950071, −8.392326159027455, −7.934184958566274, −7.758713451987143, −6.783303074410239, −6.470901794056194, −6.141633977232043, −5.396819790721056, −5.003267023820949, −4.395962162871653, −3.991101364202546, −3.442631503684940, −2.782466580433182, −2.240780488176367, −1.836524973554575, −1.170156565346212, −0.5831138022415855,
0.5831138022415855, 1.170156565346212, 1.836524973554575, 2.240780488176367, 2.782466580433182, 3.442631503684940, 3.991101364202546, 4.395962162871653, 5.003267023820949, 5.396819790721056, 6.141633977232043, 6.470901794056194, 6.783303074410239, 7.758713451987143, 7.934184958566274, 8.392326159027455, 8.689254445950071, 9.295466095715192, 9.817173583118609, 10.14126961978609, 10.77469587467845, 10.97931320665268, 11.53328147007749, 12.09391055306701, 12.45447565786807
