| L(s) = 1 | + 3-s + 2·5-s + 9-s − 13-s + 2·15-s + 2·17-s − 6·19-s + 8·23-s − 25-s + 27-s − 4·29-s + 4·31-s − 6·37-s − 39-s + 8·43-s + 2·45-s + 2·51-s − 6·57-s − 8·59-s − 10·61-s − 2·65-s + 14·67-s + 8·69-s + 8·71-s + 2·73-s − 75-s − 14·79-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s + 1/3·9-s − 0.277·13-s + 0.516·15-s + 0.485·17-s − 1.37·19-s + 1.66·23-s − 1/5·25-s + 0.192·27-s − 0.742·29-s + 0.718·31-s − 0.986·37-s − 0.160·39-s + 1.21·43-s + 0.298·45-s + 0.280·51-s − 0.794·57-s − 1.04·59-s − 1.28·61-s − 0.248·65-s + 1.71·67-s + 0.963·69-s + 0.949·71-s + 0.234·73-s − 0.115·75-s − 1.57·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 244608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 244608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.871014031\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.871014031\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 14 T + p T^{2} \) | 1.67.ao |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 8 T + p T^{2} \) | 1.89.i |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−12.98774072099689, −12.61413213600581, −12.04718487625880, −11.42823617852452, −10.92125983146220, −10.45205551791337, −10.14835194978194, −9.415817759288430, −9.278989616312781, −8.725502041739071, −8.274073091300645, −7.661543425021379, −7.264141735728030, −6.613197295526763, −6.331056883424064, −5.579900886280902, −5.302602919596230, −4.545526295558508, −4.202552697007329, −3.400517809205528, −2.981230137599111, −2.336145669548294, −1.902605567451456, −1.296787830696431, −0.5068026819100819,
0.5068026819100819, 1.296787830696431, 1.902605567451456, 2.336145669548294, 2.981230137599111, 3.400517809205528, 4.202552697007329, 4.545526295558508, 5.302602919596230, 5.579900886280902, 6.331056883424064, 6.613197295526763, 7.264141735728030, 7.661543425021379, 8.274073091300645, 8.725502041739071, 9.278989616312781, 9.415817759288430, 10.14835194978194, 10.45205551791337, 10.92125983146220, 11.42823617852452, 12.04718487625880, 12.61413213600581, 12.98774072099689
