| L(s) = 1 | + 4·5-s − 2·13-s − 4·17-s + 6·19-s − 23-s + 11·25-s − 10·29-s − 4·31-s − 2·37-s − 6·41-s − 6·43-s + 8·47-s − 8·53-s − 4·59-s + 2·61-s − 8·65-s + 6·67-s + 14·73-s − 10·79-s + 16·83-s − 16·85-s + 24·95-s + 18·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 0.554·13-s − 0.970·17-s + 1.37·19-s − 0.208·23-s + 11/5·25-s − 1.85·29-s − 0.718·31-s − 0.328·37-s − 0.937·41-s − 0.914·43-s + 1.16·47-s − 1.09·53-s − 0.520·59-s + 0.256·61-s − 0.992·65-s + 0.733·67-s + 1.63·73-s − 1.12·79-s + 1.75·83-s − 1.73·85-s + 2.46·95-s + 1.82·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.058098423\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.058098423\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 23 | \( 1 + T \) | |
| good | 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 8 T + p T^{2} \) | 1.53.i |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 6 T + p T^{2} \) | 1.67.ag |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 18 T + p T^{2} \) | 1.97.as |
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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
−13.92856670112047, −13.51912048577018, −13.00750638743380, −12.76841690562347, −11.95917541585409, −11.51120640184099, −10.86922838595743, −10.44609273992618, −9.876520291205623, −9.416010245013478, −9.176271639201107, −8.624508602502173, −7.735854719413979, −7.342147116595284, −6.649796268647047, −6.294264493518924, −5.573579983449325, −5.217313166547116, −4.835811774394480, −3.842934788951523, −3.270020845582312, −2.515087863968390, −1.928480007102483, −1.588438700130540, −0.5353995667407298,
0.5353995667407298, 1.588438700130540, 1.928480007102483, 2.515087863968390, 3.270020845582312, 3.842934788951523, 4.835811774394480, 5.217313166547116, 5.573579983449325, 6.294264493518924, 6.649796268647047, 7.342147116595284, 7.735854719413979, 8.624508602502173, 9.176271639201107, 9.416010245013478, 9.876520291205623, 10.44609273992618, 10.86922838595743, 11.51120640184099, 11.95917541585409, 12.76841690562347, 13.00750638743380, 13.51912048577018, 13.92856670112047
