| L(s) = 1 | + 2-s + 4-s − 5-s + 2·7-s + 8-s − 10-s + 6·11-s + 4·13-s + 2·14-s + 16-s + 6·17-s + 4·19-s − 20-s + 6·22-s + 25-s + 4·26-s + 2·28-s + 6·29-s + 4·31-s + 32-s + 6·34-s − 2·35-s + 4·38-s − 40-s − 8·43-s + 6·44-s − 3·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.755·7-s + 0.353·8-s − 0.316·10-s + 1.80·11-s + 1.10·13-s + 0.534·14-s + 1/4·16-s + 1.45·17-s + 0.917·19-s − 0.223·20-s + 1.27·22-s + 1/5·25-s + 0.784·26-s + 0.377·28-s + 1.11·29-s + 0.718·31-s + 0.176·32-s + 1.02·34-s − 0.338·35-s + 0.648·38-s − 0.158·40-s − 1.21·43-s + 0.904·44-s − 3/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.279395909\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.279395909\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 37 | \( 1 \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−13.69603124531805, −13.17433227711801, −12.32567803618560, −12.03197476553982, −11.79467399080031, −11.28055689442088, −10.86180653112523, −10.16061583017665, −9.727633521551985, −9.088347929476367, −8.520389919019504, −8.111379170844803, −7.596561599443696, −7.011777144435767, −6.434893405113720, −6.065263172615597, −5.471247462504951, −4.772503961580788, −4.412068273099095, −3.783697270617241, −3.255550200521985, −2.940866788033962, −1.601112529440110, −1.436646559777194, −0.8004943350052762,
0.8004943350052762, 1.436646559777194, 1.601112529440110, 2.940866788033962, 3.255550200521985, 3.783697270617241, 4.412068273099095, 4.772503961580788, 5.471247462504951, 6.065263172615597, 6.434893405113720, 7.011777144435767, 7.596561599443696, 8.111379170844803, 8.520389919019504, 9.088347929476367, 9.727633521551985, 10.16061583017665, 10.86180653112523, 11.28055689442088, 11.79467399080031, 12.03197476553982, 12.32567803618560, 13.17433227711801, 13.69603124531805
