| L(s) = 1 | + 2-s + 4-s − 2·5-s + 8-s − 2·10-s − 11-s − 2·13-s + 16-s − 6·17-s + 4·19-s − 2·20-s − 22-s + 4·23-s − 25-s − 2·26-s − 2·29-s + 4·31-s + 32-s − 6·34-s − 2·37-s + 4·38-s − 2·40-s − 6·41-s − 44-s + 4·46-s − 8·47-s − 50-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.894·5-s + 0.353·8-s − 0.632·10-s − 0.301·11-s − 0.554·13-s + 1/4·16-s − 1.45·17-s + 0.917·19-s − 0.447·20-s − 0.213·22-s + 0.834·23-s − 1/5·25-s − 0.392·26-s − 0.371·29-s + 0.718·31-s + 0.176·32-s − 1.02·34-s − 0.328·37-s + 0.648·38-s − 0.316·40-s − 0.937·41-s − 0.150·44-s + 0.589·46-s − 1.16·47-s − 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.148400901\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.148400901\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 14 T + p T^{2} \) | 1.53.ao |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−7.31524169886963257395601471545, −7.17392354889171205971358987767, −6.36063463733747478696312824906, −5.41505120623238970032232815306, −4.90279695193449377134128499565, −4.19984568778577016705058800267, −3.53143401028865361989935683360, −2.75348738236178944457442835036, −1.95056468130131064241502431444, −0.60404099945041688911466731367,
0.60404099945041688911466731367, 1.95056468130131064241502431444, 2.75348738236178944457442835036, 3.53143401028865361989935683360, 4.19984568778577016705058800267, 4.90279695193449377134128499565, 5.41505120623238970032232815306, 6.36063463733747478696312824906, 7.17392354889171205971358987767, 7.31524169886963257395601471545
