| L(s) = 1 | + 2·3-s − 5-s + 2·7-s + 9-s − 2·13-s − 2·15-s − 17-s + 8·19-s + 4·21-s + 6·23-s + 25-s − 4·27-s + 6·29-s − 2·31-s − 2·35-s + 2·37-s − 4·39-s + 6·41-s − 4·43-s − 45-s − 12·47-s − 3·49-s − 2·51-s + 6·53-s + 16·57-s − 2·61-s + 2·63-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s − 0.554·13-s − 0.516·15-s − 0.242·17-s + 1.83·19-s + 0.872·21-s + 1.25·23-s + 1/5·25-s − 0.769·27-s + 1.11·29-s − 0.359·31-s − 0.338·35-s + 0.328·37-s − 0.640·39-s + 0.937·41-s − 0.609·43-s − 0.149·45-s − 1.75·47-s − 3/7·49-s − 0.280·51-s + 0.824·53-s + 2.11·57-s − 0.256·61-s + 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−13.49663950752115, −13.19665095600178, −12.59596520104404, −11.97563053430707, −11.56807301826579, −11.27626769146509, −10.63790384548972, −10.04590547171390, −9.505838005749334, −9.175083887810009, −8.632574063749338, −8.185831696972459, −7.713019039721047, −7.407224135351797, −6.880599982592705, −6.202459780155269, −5.452121680070794, −4.812670062150829, −4.752828163720900, −3.723981717870615, −3.412816333663793, −2.727715564112103, −2.458238506320094, −1.472611048496076, −1.075594708130655, 0,
1.075594708130655, 1.472611048496076, 2.458238506320094, 2.727715564112103, 3.412816333663793, 3.723981717870615, 4.752828163720900, 4.812670062150829, 5.452121680070794, 6.202459780155269, 6.880599982592705, 7.407224135351797, 7.713019039721047, 8.185831696972459, 8.632574063749338, 9.175083887810009, 9.505838005749334, 10.04590547171390, 10.63790384548972, 11.27626769146509, 11.56807301826579, 11.97563053430707, 12.59596520104404, 13.19665095600178, 13.49663950752115
