| L(s) = 1 | − 3-s + 2·7-s + 9-s − 6·11-s + 13-s + 2·17-s − 2·19-s − 2·21-s + 4·23-s − 27-s + 2·29-s − 2·31-s + 6·33-s − 2·37-s − 39-s − 6·41-s − 6·47-s − 3·49-s − 2·51-s + 2·53-s + 2·57-s − 6·59-s + 14·61-s + 2·63-s − 2·67-s − 4·69-s − 10·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.755·7-s + 1/3·9-s − 1.80·11-s + 0.277·13-s + 0.485·17-s − 0.458·19-s − 0.436·21-s + 0.834·23-s − 0.192·27-s + 0.371·29-s − 0.359·31-s + 1.04·33-s − 0.328·37-s − 0.160·39-s − 0.937·41-s − 0.875·47-s − 3/7·49-s − 0.280·51-s + 0.274·53-s + 0.264·57-s − 0.781·59-s + 1.79·61-s + 0.251·63-s − 0.244·67-s − 0.481·69-s − 1.18·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + 10 T + p T^{2} \) | 1.71.k |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 2 T + p T^{2} \) | 1.83.c |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 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
−8.142425369545890883117355555645, −7.39911338759549255213084093414, −6.67202781543872378540003070059, −5.65593542574765453730565521629, −5.16233667984109291565045018745, −4.55518123451006424874122980227, −3.39156173284761387653479895063, −2.44209251766647477179761587071, −1.36595130074054159844538728048, 0,
1.36595130074054159844538728048, 2.44209251766647477179761587071, 3.39156173284761387653479895063, 4.55518123451006424874122980227, 5.16233667984109291565045018745, 5.65593542574765453730565521629, 6.67202781543872378540003070059, 7.39911338759549255213084093414, 8.142425369545890883117355555645
