| L(s) = 1 | − 2-s + 4-s + 2·5-s − 8-s − 2·10-s + 4·11-s + 2·13-s + 16-s + 2·20-s − 4·22-s − 25-s − 2·26-s + 6·29-s − 32-s − 2·37-s − 2·40-s + 41-s + 4·43-s + 4·44-s − 7·49-s + 50-s + 2·52-s + 6·53-s + 8·55-s − 6·58-s − 12·59-s + 6·61-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.353·8-s − 0.632·10-s + 1.20·11-s + 0.554·13-s + 1/4·16-s + 0.447·20-s − 0.852·22-s − 1/5·25-s − 0.392·26-s + 1.11·29-s − 0.176·32-s − 0.328·37-s − 0.316·40-s + 0.156·41-s + 0.609·43-s + 0.603·44-s − 49-s + 0.141·50-s + 0.277·52-s + 0.824·53-s + 1.07·55-s − 0.787·58-s − 1.56·59-s + 0.768·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 213282 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 213282 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 3 | \( 1 \) | |
| 17 | \( 1 \) | |
| 41 | \( 1 - T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 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
−13.30416606412500, −12.70187188908746, −12.17707207929236, −11.85706461923533, −11.23030223645031, −10.90664657171121, −10.29775711661173, −9.901957851976622, −9.485434777386435, −8.996784725616432, −8.694795930006361, −8.093813523077270, −7.601133559906347, −6.960054123990043, −6.416959035653135, −6.252145893086363, −5.669049388526883, −5.031913484391563, −4.445991129556355, −3.727059786888443, −3.349209940954718, −2.467990364816385, −2.108228289249576, −1.249840857750765, −1.093099780622058, 0,
1.093099780622058, 1.249840857750765, 2.108228289249576, 2.467990364816385, 3.349209940954718, 3.727059786888443, 4.445991129556355, 5.031913484391563, 5.669049388526883, 6.252145893086363, 6.416959035653135, 6.960054123990043, 7.601133559906347, 8.093813523077270, 8.694795930006361, 8.996784725616432, 9.485434777386435, 9.901957851976622, 10.29775711661173, 10.90664657171121, 11.23030223645031, 11.85706461923533, 12.17707207929236, 12.70187188908746, 13.30416606412500
