| L(s) = 1 | − 3-s − 2·7-s + 9-s + 6·13-s − 4·17-s + 2·19-s + 2·21-s + 23-s − 5·25-s − 27-s − 2·29-s − 4·31-s + 2·37-s − 6·39-s − 2·41-s + 10·43-s − 3·49-s + 4·51-s − 12·53-s − 2·57-s + 12·59-s + 6·61-s − 2·63-s + 10·67-s − 69-s − 8·71-s + 14·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.755·7-s + 1/3·9-s + 1.66·13-s − 0.970·17-s + 0.458·19-s + 0.436·21-s + 0.208·23-s − 25-s − 0.192·27-s − 0.371·29-s − 0.718·31-s + 0.328·37-s − 0.960·39-s − 0.312·41-s + 1.52·43-s − 3/7·49-s + 0.560·51-s − 1.64·53-s − 0.264·57-s + 1.56·59-s + 0.768·61-s − 0.251·63-s + 1.22·67-s − 0.120·69-s − 0.949·71-s + 1.63·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 133584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.450790499\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.450790499\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 16 T + p T^{2} \) | 1.89.q |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−13.39909609005199, −12.93155801449486, −12.63606104436912, −12.01398526566119, −11.31458310704307, −11.11402802587390, −10.80478021721892, −10.03435962848649, −9.528013122648264, −9.254886711172723, −8.580413696899046, −8.090277607647917, −7.545238554245765, −6.751174521953755, −6.612080963011150, −5.938827194671868, −5.611985865345653, −4.982068150462387, −4.210378280864081, −3.763408636326702, −3.372845667060870, −2.498006565697379, −1.861910921495054, −1.111802492525105, −0.4153496253762989,
0.4153496253762989, 1.111802492525105, 1.861910921495054, 2.498006565697379, 3.372845667060870, 3.763408636326702, 4.210378280864081, 4.982068150462387, 5.611985865345653, 5.938827194671868, 6.612080963011150, 6.751174521953755, 7.545238554245765, 8.090277607647917, 8.580413696899046, 9.254886711172723, 9.528013122648264, 10.03435962848649, 10.80478021721892, 11.11402802587390, 11.31458310704307, 12.01398526566119, 12.63606104436912, 12.93155801449486, 13.39909609005199
