| L(s) = 1 | + 3-s + 2·7-s − 2·9-s + 11-s + 4·13-s − 6·17-s − 8·19-s + 2·21-s − 3·23-s − 5·27-s − 5·31-s + 33-s + 37-s + 4·39-s − 10·43-s − 3·49-s − 6·51-s + 6·53-s − 8·57-s − 3·59-s − 4·61-s − 4·63-s − 67-s − 3·69-s − 15·71-s + 4·73-s + 2·77-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.755·7-s − 2/3·9-s + 0.301·11-s + 1.10·13-s − 1.45·17-s − 1.83·19-s + 0.436·21-s − 0.625·23-s − 0.962·27-s − 0.898·31-s + 0.174·33-s + 0.164·37-s + 0.640·39-s − 1.52·43-s − 3/7·49-s − 0.840·51-s + 0.824·53-s − 1.05·57-s − 0.390·59-s − 0.512·61-s − 0.503·63-s − 0.122·67-s − 0.361·69-s − 1.78·71-s + 0.468·73-s + 0.227·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 - T + p T^{2} \) | 1.37.ab |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 + T + p T^{2} \) | 1.67.b |
| 71 | \( 1 + 15 T + p T^{2} \) | 1.71.p |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + 2 T + p T^{2} \) | 1.79.c |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 9 T + p T^{2} \) | 1.89.j |
| 97 | \( 1 - 7 T + p T^{2} \) | 1.97.ah |
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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
−8.270291326594430363325745498073, −7.38829777934131723608792260408, −6.39194712026754517361415727782, −6.00643085529535229889961932546, −4.86824029564301859859497168572, −4.15869750405124088599031875635, −3.43029130494338377492333953612, −2.27940290108639207983747311479, −1.70082535066997228503423935134, 0,
1.70082535066997228503423935134, 2.27940290108639207983747311479, 3.43029130494338377492333953612, 4.15869750405124088599031875635, 4.86824029564301859859497168572, 6.00643085529535229889961932546, 6.39194712026754517361415727782, 7.38829777934131723608792260408, 8.270291326594430363325745498073
