| L(s) = 1 | + 3-s − 5-s + 9-s − 2·13-s − 15-s + 6·17-s − 4·19-s + 25-s + 27-s + 6·29-s − 8·31-s − 6·37-s − 2·39-s − 10·41-s − 4·43-s − 45-s − 8·47-s − 7·49-s + 6·51-s + 10·53-s − 4·57-s + 12·59-s + 6·61-s + 2·65-s + 4·67-s + 14·73-s + 75-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.554·13-s − 0.258·15-s + 1.45·17-s − 0.917·19-s + 1/5·25-s + 0.192·27-s + 1.11·29-s − 1.43·31-s − 0.986·37-s − 0.320·39-s − 1.56·41-s − 0.609·43-s − 0.149·45-s − 1.16·47-s − 49-s + 0.840·51-s + 1.37·53-s − 0.529·57-s + 1.56·59-s + 0.768·61-s + 0.248·65-s + 0.488·67-s + 1.63·73-s + 0.115·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 - T \) | |
| 5 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 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
−13.82556598821702, −13.46103202423126, −12.68705629035348, −12.52671184631299, −11.97623179419313, −11.48886551804673, −10.90863639651064, −10.32527120700147, −9.842972559172431, −9.649062932223242, −8.671444576689166, −8.432117051912799, −8.077131436456311, −7.371340036854000, −6.888587559855428, −6.584534617222940, −5.629586505905143, −5.193799616305921, −4.715649807024747, −3.913961566543526, −3.496055722812906, −3.056390323628194, −2.207091667432939, −1.733569856380140, −0.8634571254585899, 0,
0.8634571254585899, 1.733569856380140, 2.207091667432939, 3.056390323628194, 3.496055722812906, 3.913961566543526, 4.715649807024747, 5.193799616305921, 5.629586505905143, 6.584534617222940, 6.888587559855428, 7.371340036854000, 8.077131436456311, 8.432117051912799, 8.671444576689166, 9.649062932223242, 9.842972559172431, 10.32527120700147, 10.90863639651064, 11.48886551804673, 11.97623179419313, 12.52671184631299, 12.68705629035348, 13.46103202423126, 13.82556598821702
