| L(s) = 1 | + 3-s − 4·7-s + 9-s − 2·13-s + 2·17-s + 4·19-s − 4·21-s + 27-s + 6·29-s − 4·31-s + 2·37-s − 2·39-s + 6·41-s + 4·43-s − 8·47-s + 9·49-s + 2·51-s + 6·53-s + 4·57-s − 12·59-s + 10·61-s − 4·63-s + 4·67-s + 10·73-s + 12·79-s + 81-s − 12·83-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.51·7-s + 1/3·9-s − 0.554·13-s + 0.485·17-s + 0.917·19-s − 0.872·21-s + 0.192·27-s + 1.11·29-s − 0.718·31-s + 0.328·37-s − 0.320·39-s + 0.937·41-s + 0.609·43-s − 1.16·47-s + 9/7·49-s + 0.280·51-s + 0.824·53-s + 0.529·57-s − 1.56·59-s + 1.28·61-s − 0.503·63-s + 0.488·67-s + 1.17·73-s + 1.35·79-s + 1/9·81-s − 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 290400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 290400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.862146342\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.862146342\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−12.75559424435115, −12.35339482343133, −12.00417654607402, −11.28424093652467, −10.90937185027509, −10.13078013342342, −9.867564736647519, −9.657595788282227, −9.023533776157123, −8.719400210788668, −7.993432459290606, −7.527764762782035, −7.232071190813640, −6.520382932263042, −6.303743071168682, −5.606736192871283, −5.136510084193484, −4.515366591464207, −3.895770837841047, −3.374180752788817, −3.030488952417349, −2.503582124332987, −1.906702884599222, −0.9741999198448661, −0.5057253539329782,
0.5057253539329782, 0.9741999198448661, 1.906702884599222, 2.503582124332987, 3.030488952417349, 3.374180752788817, 3.895770837841047, 4.515366591464207, 5.136510084193484, 5.606736192871283, 6.303743071168682, 6.520382932263042, 7.232071190813640, 7.527764762782035, 7.993432459290606, 8.719400210788668, 9.023533776157123, 9.657595788282227, 9.867564736647519, 10.13078013342342, 10.90937185027509, 11.28424093652467, 12.00417654607402, 12.35339482343133, 12.75559424435115
