| L(s) = 1 | + 2·2-s + 2·4-s − 2·7-s + 4·13-s − 4·14-s − 4·16-s + 2·17-s − 23-s + 8·26-s − 4·28-s + 7·31-s − 8·32-s + 4·34-s − 3·37-s − 8·41-s − 6·43-s − 2·46-s + 8·47-s − 3·49-s + 8·52-s − 6·53-s − 5·59-s − 12·61-s + 14·62-s − 8·64-s + 7·67-s + 4·68-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s − 0.755·7-s + 1.10·13-s − 1.06·14-s − 16-s + 0.485·17-s − 0.208·23-s + 1.56·26-s − 0.755·28-s + 1.25·31-s − 1.41·32-s + 0.685·34-s − 0.493·37-s − 1.24·41-s − 0.914·43-s − 0.294·46-s + 1.16·47-s − 3/7·49-s + 1.10·52-s − 0.824·53-s − 0.650·59-s − 1.53·61-s + 1.77·62-s − 64-s + 0.855·67-s + 0.485·68-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 2 | \( 1 - p T + p T^{2} \) | 1.2.ac |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 7 T + p T^{2} \) | 1.31.ah |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 5 T + p T^{2} \) | 1.59.f |
| 61 | \( 1 + 12 T + p T^{2} \) | 1.61.m |
| 67 | \( 1 - 7 T + p T^{2} \) | 1.67.ah |
| 71 | \( 1 - 3 T + p T^{2} \) | 1.71.ad |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 97 | \( 1 - 7 T + p T^{2} \) | 1.97.ah |
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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
−15.33940689632482, −15.14361816389971, −14.16131652726989, −13.87032803173071, −13.53248828748555, −12.88268381343334, −12.47648635038659, −11.93342113904611, −11.48273863973369, −10.77666910539994, −10.24618270219260, −9.566832397593153, −8.999499320603777, −8.345209607147577, −7.744963111966630, −6.785311432966563, −6.452043845696772, −5.984408332162412, −5.291666276110636, −4.740173888320993, −4.012224777154048, −3.394929907785727, −3.084086947354049, −2.163685117290156, −1.210092103040835, 0,
1.210092103040835, 2.163685117290156, 3.084086947354049, 3.394929907785727, 4.012224777154048, 4.740173888320993, 5.291666276110636, 5.984408332162412, 6.452043845696772, 6.785311432966563, 7.744963111966630, 8.345209607147577, 8.999499320603777, 9.566832397593153, 10.24618270219260, 10.77666910539994, 11.48273863973369, 11.93342113904611, 12.47648635038659, 12.88268381343334, 13.53248828748555, 13.87032803173071, 14.16131652726989, 15.14361816389971, 15.33940689632482
