| L(s) = 1 | + 3-s + 2·7-s + 9-s − 6·17-s + 2·19-s + 2·21-s + 23-s − 5·25-s + 27-s + 2·29-s − 10·31-s + 2·37-s − 8·41-s + 12·43-s + 4·47-s − 3·49-s − 6·51-s + 6·53-s + 2·57-s + 8·59-s − 6·61-s + 2·63-s − 2·67-s + 69-s + 2·73-s − 5·75-s + 81-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.755·7-s + 1/3·9-s − 1.45·17-s + 0.458·19-s + 0.436·21-s + 0.208·23-s − 25-s + 0.192·27-s + 0.371·29-s − 1.79·31-s + 0.328·37-s − 1.24·41-s + 1.82·43-s + 0.583·47-s − 3/7·49-s − 0.840·51-s + 0.824·53-s + 0.264·57-s + 1.04·59-s − 0.768·61-s + 0.251·63-s − 0.244·67-s + 0.120·69-s + 0.234·73-s − 0.577·75-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 186576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 186576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.656509981\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.656509981\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−13.26501810294111, −12.70996469970017, −12.20224236672875, −11.69913659726023, −11.09107517451986, −10.96935527147688, −10.29963347546806, −9.740873339111511, −9.225400654415705, −8.854204143997524, −8.427855351867390, −7.828250592508917, −7.441758893592328, −6.957609241410618, −6.439020760521884, −5.673839500112431, −5.347327913568049, −4.647236281454597, −4.142028542403524, −3.776701865564216, −2.999651619519361, −2.384508180636764, −1.916453317890962, −1.350736900297906, −0.4337363136578870,
0.4337363136578870, 1.350736900297906, 1.916453317890962, 2.384508180636764, 2.999651619519361, 3.776701865564216, 4.142028542403524, 4.647236281454597, 5.347327913568049, 5.673839500112431, 6.439020760521884, 6.957609241410618, 7.441758893592328, 7.828250592508917, 8.427855351867390, 8.854204143997524, 9.225400654415705, 9.740873339111511, 10.29963347546806, 10.96935527147688, 11.09107517451986, 11.69913659726023, 12.20224236672875, 12.70996469970017, 13.26501810294111
