| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 8-s + 9-s − 10-s + 4·11-s − 12-s − 15-s + 16-s − 18-s − 4·19-s + 20-s − 4·22-s − 2·23-s + 24-s + 25-s − 27-s + 8·29-s + 30-s + 6·31-s − 32-s − 4·33-s + 36-s + 8·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.20·11-s − 0.288·12-s − 0.258·15-s + 1/4·16-s − 0.235·18-s − 0.917·19-s + 0.223·20-s − 0.852·22-s − 0.417·23-s + 0.204·24-s + 1/5·25-s − 0.192·27-s + 1.48·29-s + 0.182·30-s + 1.07·31-s − 0.176·32-s − 0.696·33-s + 1/6·36-s + 1.31·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 248430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 248430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.430352953\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.430352953\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−12.65913680121588, −12.37552233250421, −11.79994935704034, −11.47099850350822, −10.95069083515995, −10.57149956864436, −9.981942684724068, −9.601262302093737, −9.341905900235114, −8.576545633579690, −8.248799808117236, −7.843890891354239, −6.981897139848574, −6.633640675947668, −6.362968064021751, −5.873132610191461, −5.278668150678397, −4.550090110675092, −4.217348186094518, −3.599934634088035, −2.758934568062229, −2.308061950818068, −1.665182217704776, −0.9159881374204732, −0.6376691046343329,
0.6376691046343329, 0.9159881374204732, 1.665182217704776, 2.308061950818068, 2.758934568062229, 3.599934634088035, 4.217348186094518, 4.550090110675092, 5.278668150678397, 5.873132610191461, 6.362968064021751, 6.633640675947668, 6.981897139848574, 7.843890891354239, 8.248799808117236, 8.576545633579690, 9.341905900235114, 9.601262302093737, 9.981942684724068, 10.57149956864436, 10.95069083515995, 11.47099850350822, 11.79994935704034, 12.37552233250421, 12.65913680121588
