| L(s) = 1 | + 2-s + 4-s − 5-s + 8-s − 10-s − 13-s + 16-s − 20-s − 4·23-s + 25-s − 26-s − 10·29-s + 32-s + 6·37-s − 40-s + 2·41-s − 4·43-s − 4·46-s − 7·49-s + 50-s − 52-s + 6·53-s − 10·58-s − 6·61-s + 64-s + 65-s + 4·67-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.353·8-s − 0.316·10-s − 0.277·13-s + 1/4·16-s − 0.223·20-s − 0.834·23-s + 1/5·25-s − 0.196·26-s − 1.85·29-s + 0.176·32-s + 0.986·37-s − 0.158·40-s + 0.312·41-s − 0.609·43-s − 0.589·46-s − 49-s + 0.141·50-s − 0.138·52-s + 0.824·53-s − 1.31·58-s − 0.768·61-s + 1/8·64-s + 0.124·65-s + 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 13 | \( 1 + T \) | |
| 17 | \( 1 \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 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
−12.79055918701686, −12.44403625952654, −11.87782265832764, −11.53432022751143, −11.05077084385802, −10.79497119119158, −10.00869416435134, −9.726575651272046, −9.229142911574568, −8.607689869152910, −8.075852250497370, −7.661868503103249, −7.322764640248345, −6.674809454550146, −6.278429639799706, −5.718700300872541, −5.267824078215947, −4.779463495492995, −4.167477153905908, −3.807107754799569, −3.328320593775829, −2.677325676353384, −2.104214540030010, −1.613903396058674, −0.7440092542061218, 0,
0.7440092542061218, 1.613903396058674, 2.104214540030010, 2.677325676353384, 3.328320593775829, 3.807107754799569, 4.167477153905908, 4.779463495492995, 5.267824078215947, 5.718700300872541, 6.278429639799706, 6.674809454550146, 7.322764640248345, 7.661868503103249, 8.075852250497370, 8.607689869152910, 9.229142911574568, 9.726575651272046, 10.00869416435134, 10.79497119119158, 11.05077084385802, 11.53432022751143, 11.87782265832764, 12.44403625952654, 12.79055918701686
