| L(s) = 1 | − 2-s + 4-s + 5-s − 8-s − 10-s − 5·13-s + 16-s − 2·17-s + 4·19-s + 20-s − 4·25-s + 5·26-s + 5·29-s + 4·31-s − 32-s + 2·34-s − 10·37-s − 4·38-s − 40-s + 5·41-s + 8·43-s + 4·47-s − 7·49-s + 4·50-s − 5·52-s − 11·53-s − 5·58-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.353·8-s − 0.316·10-s − 1.38·13-s + 1/4·16-s − 0.485·17-s + 0.917·19-s + 0.223·20-s − 4/5·25-s + 0.980·26-s + 0.928·29-s + 0.718·31-s − 0.176·32-s + 0.342·34-s − 1.64·37-s − 0.648·38-s − 0.158·40-s + 0.780·41-s + 1.21·43-s + 0.583·47-s − 49-s + 0.565·50-s − 0.693·52-s − 1.51·53-s − 0.656·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 23 | \( 1 \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 29 | \( 1 - 5 T + p T^{2} \) | 1.29.af |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 - 5 T + p T^{2} \) | 1.41.af |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + 11 T + p T^{2} \) | 1.53.l |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 3 T + p T^{2} \) | 1.73.ad |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 - 9 T + p T^{2} \) | 1.89.aj |
| 97 | \( 1 + 5 T + p T^{2} \) | 1.97.f |
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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
−7.50630897166255946633783876901, −6.73164363014448174207391637507, −6.17424989412051350238388298927, −5.27231578169856551120344278343, −4.75662299137631591019409137746, −3.70967662640645347716212928005, −2.73487156384656307891372064728, −2.18403841014540006703362837864, −1.16628371137790021634658001304, 0,
1.16628371137790021634658001304, 2.18403841014540006703362837864, 2.73487156384656307891372064728, 3.70967662640645347716212928005, 4.75662299137631591019409137746, 5.27231578169856551120344278343, 6.17424989412051350238388298927, 6.73164363014448174207391637507, 7.50630897166255946633783876901
