| L(s) = 1 | + 2-s + 4-s + 5-s − 7-s + 8-s + 10-s + 4·11-s − 13-s − 14-s + 16-s − 6·17-s + 20-s + 4·22-s − 4·23-s + 25-s − 26-s − 28-s + 2·29-s + 4·31-s + 32-s − 6·34-s − 35-s + 2·37-s + 40-s + 2·41-s + 8·43-s + 4·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.377·7-s + 0.353·8-s + 0.316·10-s + 1.20·11-s − 0.277·13-s − 0.267·14-s + 1/4·16-s − 1.45·17-s + 0.223·20-s + 0.852·22-s − 0.834·23-s + 1/5·25-s − 0.196·26-s − 0.188·28-s + 0.371·29-s + 0.718·31-s + 0.176·32-s − 1.02·34-s − 0.169·35-s + 0.328·37-s + 0.158·40-s + 0.312·41-s + 1.21·43-s + 0.603·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.617363012\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.617363012\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 + T \) | |
| good | 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 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
−7.56879224577358620933827898952, −6.96303177994543610539733156974, −6.20796609208853362452083631728, −6.00260411757249723331297371755, −4.88805145590232112525356613820, −4.26773433357495107548315147554, −3.67986552242223756770786941545, −2.60163190553441946798733326227, −2.04009794769523594196708630223, −0.842897524720167454598179757372,
0.842897524720167454598179757372, 2.04009794769523594196708630223, 2.60163190553441946798733326227, 3.67986552242223756770786941545, 4.26773433357495107548315147554, 4.88805145590232112525356613820, 6.00260411757249723331297371755, 6.20796609208853362452083631728, 6.96303177994543610539733156974, 7.56879224577358620933827898952
