| L(s) = 1 | − 3·5-s − 4·13-s − 6·17-s + 8·19-s + 3·23-s + 4·25-s − 5·31-s − 37-s + 10·43-s + 6·53-s + 3·59-s − 4·61-s + 12·65-s − 67-s − 15·71-s − 4·73-s − 2·79-s − 6·83-s + 18·85-s − 9·89-s − 24·95-s + 7·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 1.10·13-s − 1.45·17-s + 1.83·19-s + 0.625·23-s + 4/5·25-s − 0.898·31-s − 0.164·37-s + 1.52·43-s + 0.824·53-s + 0.390·59-s − 0.512·61-s + 1.48·65-s − 0.122·67-s − 1.78·71-s − 0.468·73-s − 0.225·79-s − 0.658·83-s + 1.95·85-s − 0.953·89-s − 2.46·95-s + 0.710·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 213444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 213444 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 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.b |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 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.ad |
| 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.e |
| 79 | \( 1 + 2 T + p T^{2} \) | 1.79.c |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 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
−13.14849111991631, −12.70884596551652, −12.18777528315213, −11.83745898424101, −11.42525766765137, −11.00443463309808, −10.58918980605246, −9.871835587650902, −9.461234976730739, −8.936326311351367, −8.581794767580622, −7.875508760445789, −7.466452012453714, −7.086074474802742, −6.891688763065918, −5.830751905403143, −5.540167929214323, −4.775878236470699, −4.453696909575306, −3.961941476717069, −3.278919323974661, −2.849921005280947, −2.241858969125582, −1.409094326629193, −0.6255365856658552, 0,
0.6255365856658552, 1.409094326629193, 2.241858969125582, 2.849921005280947, 3.278919323974661, 3.961941476717069, 4.453696909575306, 4.775878236470699, 5.540167929214323, 5.830751905403143, 6.891688763065918, 7.086074474802742, 7.466452012453714, 7.875508760445789, 8.581794767580622, 8.936326311351367, 9.461234976730739, 9.871835587650902, 10.58918980605246, 11.00443463309808, 11.42525766765137, 11.83745898424101, 12.18777528315213, 12.70884596551652, 13.14849111991631
