| L(s) = 1 | + 3-s + 2·5-s − 2·9-s + 11-s + 3·13-s + 2·15-s − 2·17-s − 4·19-s + 4·23-s − 25-s − 5·27-s − 7·29-s + 8·31-s + 33-s + 12·37-s + 3·39-s − 8·41-s − 8·43-s − 4·45-s + 10·47-s − 2·51-s + 14·53-s + 2·55-s − 4·57-s + 9·59-s − 5·61-s + 6·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s − 2/3·9-s + 0.301·11-s + 0.832·13-s + 0.516·15-s − 0.485·17-s − 0.917·19-s + 0.834·23-s − 1/5·25-s − 0.962·27-s − 1.29·29-s + 1.43·31-s + 0.174·33-s + 1.97·37-s + 0.480·39-s − 1.24·41-s − 1.21·43-s − 0.596·45-s + 1.45·47-s − 0.280·51-s + 1.92·53-s + 0.269·55-s − 0.529·57-s + 1.17·59-s − 0.640·61-s + 0.744·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.977718703\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.977718703\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 13 | \( 1 - 3 T + p T^{2} \) | 1.13.ad |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 7 T + p T^{2} \) | 1.29.h |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 12 T + p T^{2} \) | 1.37.am |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 53 | \( 1 - 14 T + p T^{2} \) | 1.53.ao |
| 59 | \( 1 - 9 T + p T^{2} \) | 1.59.aj |
| 61 | \( 1 + 5 T + p T^{2} \) | 1.61.f |
| 67 | \( 1 - 3 T + p T^{2} \) | 1.67.ad |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 - 17 T + p T^{2} \) | 1.79.ar |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 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
−7.976393109287411883837036118592, −6.98961193350023077900599954290, −6.31777606632748671066113064633, −5.83742658800599768792910896575, −5.07594837173912661879213190321, −4.11282474509209200346343655278, −3.45475243729387401578533534375, −2.47405769437409514114859322130, −2.00684921990468849997474512052, −0.813048474466193814798899274179,
0.813048474466193814798899274179, 2.00684921990468849997474512052, 2.47405769437409514114859322130, 3.45475243729387401578533534375, 4.11282474509209200346343655278, 5.07594837173912661879213190321, 5.83742658800599768792910896575, 6.31777606632748671066113064633, 6.98961193350023077900599954290, 7.976393109287411883837036118592
