| L(s) = 1 | + 5-s + 5·11-s + 3·13-s + 3·17-s + 4·19-s − 7·23-s + 25-s − 8·29-s + 3·31-s − 3·41-s + 43-s − 7·49-s − 9·53-s + 5·55-s − 8·59-s + 3·65-s − 11·67-s + 8·71-s − 4·73-s + 8·79-s − 9·83-s + 3·85-s + 4·89-s + 4·95-s + 13·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.50·11-s + 0.832·13-s + 0.727·17-s + 0.917·19-s − 1.45·23-s + 1/5·25-s − 1.48·29-s + 0.538·31-s − 0.468·41-s + 0.152·43-s − 49-s − 1.23·53-s + 0.674·55-s − 1.04·59-s + 0.372·65-s − 1.34·67-s + 0.949·71-s − 0.468·73-s + 0.900·79-s − 0.987·83-s + 0.325·85-s + 0.423·89-s + 0.410·95-s + 1.31·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 43 | \( 1 - T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 13 | \( 1 - 3 T + p T^{2} \) | 1.13.ad |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 7 T + p T^{2} \) | 1.23.h |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 - 3 T + p T^{2} \) | 1.31.ad |
| 37 | \( 1 + p T^{2} \) | 1.37.a |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 11 T + p T^{2} \) | 1.67.l |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 - 4 T + p T^{2} \) | 1.89.ae |
| 97 | \( 1 - 13 T + p T^{2} \) | 1.97.an |
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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
−13.89765135305062, −13.31621667877346, −12.89901385798581, −12.14581906387692, −11.89971682534202, −11.47975030509923, −10.85609938550077, −10.42033097906636, −9.612069174866515, −9.528671761692463, −9.071908126213887, −8.277134106667691, −7.975277341127286, −7.332369158324665, −6.742457581626438, −6.161987250710188, −5.918382731186037, −5.314642225012623, −4.580830994471992, −4.017134977669014, −3.462457866806530, −3.090881089265647, −2.042241983153534, −1.529872409818570, −1.088328204035911, 0,
1.088328204035911, 1.529872409818570, 2.042241983153534, 3.090881089265647, 3.462457866806530, 4.017134977669014, 4.580830994471992, 5.314642225012623, 5.918382731186037, 6.161987250710188, 6.742457581626438, 7.332369158324665, 7.975277341127286, 8.277134106667691, 9.071908126213887, 9.528671761692463, 9.612069174866515, 10.42033097906636, 10.85609938550077, 11.47975030509923, 11.89971682534202, 12.14581906387692, 12.89901385798581, 13.31621667877346, 13.89765135305062
