| L(s) = 1 | + 2-s − 3-s − 4-s + 5-s − 6-s + 4·7-s − 3·8-s + 9-s + 10-s + 4·11-s + 12-s + 4·14-s − 15-s − 16-s − 17-s + 18-s + 4·19-s − 20-s − 4·21-s + 4·22-s + 3·24-s + 25-s − 27-s − 4·28-s − 6·29-s − 30-s + 5·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.447·5-s − 0.408·6-s + 1.51·7-s − 1.06·8-s + 1/3·9-s + 0.316·10-s + 1.20·11-s + 0.288·12-s + 1.06·14-s − 0.258·15-s − 1/4·16-s − 0.242·17-s + 0.235·18-s + 0.917·19-s − 0.223·20-s − 0.872·21-s + 0.852·22-s + 0.612·24-s + 1/5·25-s − 0.192·27-s − 0.755·28-s − 1.11·29-s − 0.182·30-s + 0.883·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43095 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43095 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.725341845\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.725341845\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 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
−14.53029916333798, −14.20778466964587, −13.77262678600257, −13.14141657965959, −12.74181623873703, −11.96940055617879, −11.66862646084823, −11.27690486253200, −10.71701220392589, −9.891730090883354, −9.411974062283305, −8.990396639555243, −8.361301731433599, −7.694060168149505, −7.162241813127942, −6.346045278262059, −5.862740506106765, −5.379989799050203, −4.830481323271307, −4.283329537809091, −3.881097333342918, −2.990099770277320, −2.086490882229479, −1.353640609603246, −0.7119750511685691,
0.7119750511685691, 1.353640609603246, 2.086490882229479, 2.990099770277320, 3.881097333342918, 4.283329537809091, 4.830481323271307, 5.379989799050203, 5.862740506106765, 6.346045278262059, 7.162241813127942, 7.694060168149505, 8.361301731433599, 8.990396639555243, 9.411974062283305, 9.891730090883354, 10.71701220392589, 11.27690486253200, 11.66862646084823, 11.96940055617879, 12.74181623873703, 13.14141657965959, 13.77262678600257, 14.20778466964587, 14.53029916333798
