| L(s) = 1 | − 2-s + 3-s + 4-s − 3·5-s − 6-s − 7-s − 8-s + 9-s + 3·10-s + 12-s + 14-s − 3·15-s + 16-s − 18-s + 2·19-s − 3·20-s − 21-s + 3·23-s − 24-s + 4·25-s + 27-s − 28-s + 3·29-s + 3·30-s − 8·31-s − 32-s + 3·35-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.34·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.948·10-s + 0.288·12-s + 0.267·14-s − 0.774·15-s + 1/4·16-s − 0.235·18-s + 0.458·19-s − 0.670·20-s − 0.218·21-s + 0.625·23-s − 0.204·24-s + 4/5·25-s + 0.192·27-s − 0.188·28-s + 0.557·29-s + 0.547·30-s − 1.43·31-s − 0.176·32-s + 0.507·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122694 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122694 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 3 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 + 11 T + p T^{2} \) | 1.67.l |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + T + p T^{2} \) | 1.73.b |
| 79 | \( 1 + 2 T + p T^{2} \) | 1.79.c |
| 83 | \( 1 - 18 T + p T^{2} \) | 1.83.as |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 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
−13.78992341756673, −13.23697949368295, −12.76564650835855, −12.21429252431120, −11.87401099215650, −11.21312003502978, −11.04286588789533, −10.18336590828501, −9.986717226236728, −9.235571156632598, −8.831674225346358, −8.454261216603982, −7.799923778725890, −7.572606387066090, −6.972007012368255, −6.600733494795119, −5.884869573100354, −5.031972220575753, −4.684959053141307, −3.812888188083469, −3.309203682909644, −3.160975150113557, −2.158031338126329, −1.561827351068482, −0.6933143719718526, 0,
0.6933143719718526, 1.561827351068482, 2.158031338126329, 3.160975150113557, 3.309203682909644, 3.812888188083469, 4.684959053141307, 5.031972220575753, 5.884869573100354, 6.600733494795119, 6.972007012368255, 7.572606387066090, 7.799923778725890, 8.454261216603982, 8.831674225346358, 9.235571156632598, 9.986717226236728, 10.18336590828501, 11.04286588789533, 11.21312003502978, 11.87401099215650, 12.21429252431120, 12.76564650835855, 13.23697949368295, 13.78992341756673
