| L(s) = 1 | − 3·7-s − 3·9-s − 5·11-s − 2·13-s − 5·17-s − 2·19-s + 3·23-s + 7·29-s − 3·31-s − 11·37-s − 5·41-s − 6·43-s − 4·47-s + 2·49-s + 10·53-s − 12·59-s + 14·61-s + 9·63-s − 2·67-s + 12·71-s + 4·73-s + 15·77-s − 14·79-s + 9·81-s + 83-s − 6·89-s + 6·91-s + ⋯ |
| L(s) = 1 | − 1.13·7-s − 9-s − 1.50·11-s − 0.554·13-s − 1.21·17-s − 0.458·19-s + 0.625·23-s + 1.29·29-s − 0.538·31-s − 1.80·37-s − 0.780·41-s − 0.914·43-s − 0.583·47-s + 2/7·49-s + 1.37·53-s − 1.56·59-s + 1.79·61-s + 1.13·63-s − 0.244·67-s + 1.42·71-s + 0.468·73-s + 1.70·77-s − 1.57·79-s + 81-s + 0.109·83-s − 0.635·89-s + 0.628·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 132800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 132800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 83 | \( 1 - T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 5 T + p T^{2} \) | 1.17.f |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 - 7 T + p T^{2} \) | 1.29.ah |
| 31 | \( 1 + 3 T + p T^{2} \) | 1.31.d |
| 37 | \( 1 + 11 T + p T^{2} \) | 1.37.l |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 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
−13.64773423462952, −13.26097064225265, −12.67371182493505, −12.47844312374617, −11.75258202836231, −11.32206638702970, −10.67114811737805, −10.38630758002892, −9.934217858496544, −9.309141544805464, −8.772873127486843, −8.380619590291194, −7.987317317159561, −7.065905765377073, −6.825707900851775, −6.380136130601387, −5.612482887461485, −5.177575634791485, −4.823216205560250, −3.969200598475438, −3.309207300696133, −2.811642957997602, −2.454526522801469, −1.730543120547165, −0.4569066742039636, 0,
0.4569066742039636, 1.730543120547165, 2.454526522801469, 2.811642957997602, 3.309207300696133, 3.969200598475438, 4.823216205560250, 5.177575634791485, 5.612482887461485, 6.380136130601387, 6.825707900851775, 7.065905765377073, 7.987317317159561, 8.380619590291194, 8.772873127486843, 9.309141544805464, 9.934217858496544, 10.38630758002892, 10.67114811737805, 11.32206638702970, 11.75258202836231, 12.47844312374617, 12.67371182493505, 13.26097064225265, 13.64773423462952
