| L(s) = 1 | − 2-s − 4-s + 7-s + 3·8-s + 13-s − 14-s − 16-s + 2·17-s − 4·19-s + 6·23-s − 26-s − 28-s + 4·29-s − 5·32-s − 2·34-s + 10·37-s + 4·38-s − 12·41-s + 4·43-s − 6·46-s − 10·47-s + 49-s − 52-s − 12·53-s + 3·56-s − 4·58-s + 14·59-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.377·7-s + 1.06·8-s + 0.277·13-s − 0.267·14-s − 1/4·16-s + 0.485·17-s − 0.917·19-s + 1.25·23-s − 0.196·26-s − 0.188·28-s + 0.742·29-s − 0.883·32-s − 0.342·34-s + 1.64·37-s + 0.648·38-s − 1.87·41-s + 0.609·43-s − 0.884·46-s − 1.45·47-s + 1/7·49-s − 0.138·52-s − 1.64·53-s + 0.400·56-s − 0.525·58-s + 1.82·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 13 | \( 1 - T \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 10 T + p T^{2} \) | 1.47.k |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 - 14 T + p T^{2} \) | 1.59.ao |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−16.10348354349758, −15.33573977090206, −14.69250581372500, −14.44601531070245, −13.66784431437529, −13.11010802636398, −12.83315144118011, −11.99406816116850, −11.34305401691147, −10.86992811006375, −10.26056087050313, −9.798796528951929, −9.129067200444882, −8.641139430410505, −8.110501842696626, −7.669357232149782, −6.855057617688869, −6.309656616548800, −5.419978465546264, −4.799021131302345, −4.341759016062673, −3.483862005169213, −2.707322941843403, −1.650440371194251, −1.054638906644435, 0,
1.054638906644435, 1.650440371194251, 2.707322941843403, 3.483862005169213, 4.341759016062673, 4.799021131302345, 5.419978465546264, 6.309656616548800, 6.855057617688869, 7.669357232149782, 8.110501842696626, 8.641139430410505, 9.129067200444882, 9.798796528951929, 10.26056087050313, 10.86992811006375, 11.34305401691147, 11.99406816116850, 12.83315144118011, 13.11010802636398, 13.66784431437529, 14.44601531070245, 14.69250581372500, 15.33573977090206, 16.10348354349758
