| L(s) = 1 | − 3-s + 7-s + 9-s − 11-s + 2·13-s + 4·17-s + 6·19-s − 21-s − 4·23-s − 5·25-s − 27-s + 2·29-s + 10·31-s + 33-s + 6·37-s − 2·39-s − 12·41-s + 12·43-s − 6·47-s + 49-s − 4·51-s + 6·53-s − 6·57-s − 6·61-s + 63-s + 4·67-s + 4·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.301·11-s + 0.554·13-s + 0.970·17-s + 1.37·19-s − 0.218·21-s − 0.834·23-s − 25-s − 0.192·27-s + 0.371·29-s + 1.79·31-s + 0.174·33-s + 0.986·37-s − 0.320·39-s − 1.87·41-s + 1.82·43-s − 0.875·47-s + 1/7·49-s − 0.560·51-s + 0.824·53-s − 0.794·57-s − 0.768·61-s + 0.125·63-s + 0.488·67-s + 0.481·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.064088911\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.064088911\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 3 | \( 1 + T \) | |
| 7 | \( 1 - T \) | |
| 11 | \( 1 + T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 - 2 T + p T^{2} \) | 1.83.ac |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−16.22209569415424, −15.50367729572972, −15.18276075829249, −14.26654528525281, −13.74231026533259, −13.50639295484384, −12.55473213156080, −11.94593529087888, −11.73976886768592, −11.06348979922569, −10.30620378486494, −9.933958623045640, −9.367883671097311, −8.427152410109727, −7.865286102090435, −7.520595719147124, −6.527672569301408, −6.034054624795884, −5.391819897061849, −4.829521198474102, −4.020143909294133, −3.326495175911349, −2.446982491791807, −1.421606706019449, −0.7094608938093527,
0.7094608938093527, 1.421606706019449, 2.446982491791807, 3.326495175911349, 4.020143909294133, 4.829521198474102, 5.391819897061849, 6.034054624795884, 6.527672569301408, 7.520595719147124, 7.865286102090435, 8.427152410109727, 9.367883671097311, 9.933958623045640, 10.30620378486494, 11.06348979922569, 11.73976886768592, 11.94593529087888, 12.55473213156080, 13.50639295484384, 13.74231026533259, 14.26654528525281, 15.18276075829249, 15.50367729572972, 16.22209569415424
