| L(s) = 1 | − 2-s + 4-s − 5-s − 7-s − 8-s + 10-s + 4·11-s − 13-s + 14-s + 16-s − 6·19-s − 20-s − 4·22-s + 2·23-s + 25-s + 26-s − 28-s − 6·29-s − 8·31-s − 32-s + 35-s − 6·37-s + 6·38-s + 40-s + 8·41-s + 4·43-s + 4·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.377·7-s − 0.353·8-s + 0.316·10-s + 1.20·11-s − 0.277·13-s + 0.267·14-s + 1/4·16-s − 1.37·19-s − 0.223·20-s − 0.852·22-s + 0.417·23-s + 1/5·25-s + 0.196·26-s − 0.188·28-s − 1.11·29-s − 1.43·31-s − 0.176·32-s + 0.169·35-s − 0.986·37-s + 0.973·38-s + 0.158·40-s + 1.24·41-s + 0.609·43-s + 0.603·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9606720710\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9606720710\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 + T \) | |
| good | 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 16 T + p T^{2} \) | 1.89.aq |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−7.66712251879193662854477734964, −7.30658446311593259469391752126, −6.51760597517957713283382859499, −6.00769507545392160783012673577, −5.05451277383289200014864382360, −4.00335072733598328521015264847, −3.63950900386493333419831036047, −2.48653125652622753566257427669, −1.68517807872807561215715208746, −0.53558312704812234777508614905,
0.53558312704812234777508614905, 1.68517807872807561215715208746, 2.48653125652622753566257427669, 3.63950900386493333419831036047, 4.00335072733598328521015264847, 5.05451277383289200014864382360, 6.00769507545392160783012673577, 6.51760597517957713283382859499, 7.30658446311593259469391752126, 7.66712251879193662854477734964
