| L(s) = 1 | + 3-s + 9-s − 2·11-s + 2·13-s + 4·19-s − 6·23-s + 27-s + 10·29-s − 8·31-s − 2·33-s + 10·37-s + 2·39-s + 4·41-s + 8·43-s + 4·47-s + 10·53-s + 4·57-s − 8·59-s − 6·61-s − 4·67-s − 6·69-s − 14·71-s + 6·73-s − 4·79-s + 81-s − 12·83-s + 10·87-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s − 0.603·11-s + 0.554·13-s + 0.917·19-s − 1.25·23-s + 0.192·27-s + 1.85·29-s − 1.43·31-s − 0.348·33-s + 1.64·37-s + 0.320·39-s + 0.624·41-s + 1.21·43-s + 0.583·47-s + 1.37·53-s + 0.529·57-s − 1.04·59-s − 0.768·61-s − 0.488·67-s − 0.722·69-s − 1.66·71-s + 0.702·73-s − 0.450·79-s + 1/9·81-s − 1.31·83-s + 1.07·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 14 T + p T^{2} \) | 1.71.o |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 4 T + p T^{2} \) | 1.89.e |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−13.26114541233897, −12.73246360358583, −12.09031145122279, −12.00391913259449, −11.14182567095419, −10.85501225646600, −10.30080404360609, −9.867144457183262, −9.420399706955613, −8.850751969866307, −8.533358021796911, −7.842308368571788, −7.595220259359646, −7.175836707823771, −6.394637894330256, −5.913907716323804, −5.589405517225397, −4.847387130380978, −4.213446028856804, −3.985973582297433, −3.152145473355105, −2.722442253256916, −2.278903002271396, −1.415788342872208, −0.9417965102723276, 0,
0.9417965102723276, 1.415788342872208, 2.278903002271396, 2.722442253256916, 3.152145473355105, 3.985973582297433, 4.213446028856804, 4.847387130380978, 5.589405517225397, 5.913907716323804, 6.394637894330256, 7.175836707823771, 7.595220259359646, 7.842308368571788, 8.533358021796911, 8.850751969866307, 9.420399706955613, 9.867144457183262, 10.30080404360609, 10.85501225646600, 11.14182567095419, 12.00391913259449, 12.09031145122279, 12.73246360358583, 13.26114541233897
