| L(s) = 1 | + 2-s + 4-s + 8-s + 16-s + 2·17-s − 2·19-s − 6·23-s − 2·29-s + 32-s + 2·34-s + 6·37-s − 2·38-s − 2·41-s − 2·43-s − 6·46-s + 2·47-s − 7·49-s − 2·53-s − 2·58-s + 8·59-s − 4·61-s + 64-s + 4·67-s + 2·68-s − 2·71-s − 10·73-s + 6·74-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s + 1/4·16-s + 0.485·17-s − 0.458·19-s − 1.25·23-s − 0.371·29-s + 0.176·32-s + 0.342·34-s + 0.986·37-s − 0.324·38-s − 0.312·41-s − 0.304·43-s − 0.884·46-s + 0.291·47-s − 49-s − 0.274·53-s − 0.262·58-s + 1.04·59-s − 0.512·61-s + 1/8·64-s + 0.488·67-s + 0.242·68-s − 0.237·71-s − 1.17·73-s + 0.697·74-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 18 T + p T^{2} \) | 1.97.as |
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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
−14.67307611466692, −14.21928540259210, −13.64667392528238, −13.11805893987757, −12.76384845645436, −12.13035503433242, −11.68819694922777, −11.27981543475465, −10.56633539677463, −10.14516951049801, −9.626635892069517, −8.981739327602330, −8.275984310061694, −7.858729398507129, −7.287624400343183, −6.650160123681773, −6.009650075564761, −5.757689375931660, −4.870566685287225, −4.502874157194706, −3.719103572993444, −3.334960900085937, −2.442277475916747, −1.937704628656255, −1.076743351307710, 0,
1.076743351307710, 1.937704628656255, 2.442277475916747, 3.334960900085937, 3.719103572993444, 4.502874157194706, 4.870566685287225, 5.757689375931660, 6.009650075564761, 6.650160123681773, 7.287624400343183, 7.858729398507129, 8.275984310061694, 8.981739327602330, 9.626635892069517, 10.14516951049801, 10.56633539677463, 11.27981543475465, 11.68819694922777, 12.13035503433242, 12.76384845645436, 13.11805893987757, 13.64667392528238, 14.21928540259210, 14.67307611466692
