| L(s) = 1 | + 2-s + 3-s − 4-s + 6-s − 3·8-s + 9-s − 12-s − 16-s − 4·17-s + 18-s + 4·19-s + 4·23-s − 3·24-s + 27-s + 29-s + 8·31-s + 5·32-s − 4·34-s − 36-s + 4·38-s + 10·41-s + 4·43-s + 4·46-s + 12·47-s − 48-s − 4·51-s + 12·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.408·6-s − 1.06·8-s + 1/3·9-s − 0.288·12-s − 1/4·16-s − 0.970·17-s + 0.235·18-s + 0.917·19-s + 0.834·23-s − 0.612·24-s + 0.192·27-s + 0.185·29-s + 1.43·31-s + 0.883·32-s − 0.685·34-s − 1/6·36-s + 0.648·38-s + 1.56·41-s + 0.609·43-s + 0.589·46-s + 1.75·47-s − 0.144·48-s − 0.560·51-s + 1.64·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 106575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 106575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.332685247\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.332685247\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| 29 | \( 1 - T \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + p T^{2} \) | 1.37.a |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−13.75859718487095, −13.26258501907852, −12.88579949190824, −12.37795908332631, −11.81766140691299, −11.45885553441686, −10.70543031735677, −10.25723095350993, −9.651814356597837, −9.062140719932345, −8.933372092547730, −8.295293028979559, −7.710512967086357, −7.131156035814897, −6.650179971449491, −5.919660105051465, −5.524257398984363, −4.883476020503249, −4.224199213837797, −4.103223790788534, −3.224215688201607, −2.702429248358467, −2.318334027240480, −1.132875633988376, −0.6346344195884812,
0.6346344195884812, 1.132875633988376, 2.318334027240480, 2.702429248358467, 3.224215688201607, 4.103223790788534, 4.224199213837797, 4.883476020503249, 5.524257398984363, 5.919660105051465, 6.650179971449491, 7.131156035814897, 7.710512967086357, 8.295293028979559, 8.933372092547730, 9.062140719932345, 9.651814356597837, 10.25723095350993, 10.70543031735677, 11.45885553441686, 11.81766140691299, 12.37795908332631, 12.88579949190824, 13.26258501907852, 13.75859718487095
