| L(s) = 1 | − 2-s + 4-s + 5-s − 7-s − 8-s − 10-s + 13-s + 14-s + 16-s − 4·17-s − 6·19-s + 20-s + 25-s − 26-s − 28-s + 6·29-s + 4·31-s − 32-s + 4·34-s − 35-s − 8·37-s + 6·38-s − 40-s + 2·41-s + 6·43-s + 10·47-s + 49-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 + 0.277·13-s + 0.267·14-s + 1/4·16-s − 0.970·17-s − 1.37·19-s + 0.223·20-s + 1/5·25-s − 0.196·26-s − 0.188·28-s + 1.11·29-s + 0.718·31-s − 0.176·32-s + 0.685·34-s − 0.169·35-s − 1.31·37-s + 0.973·38-s − 0.158·40-s + 0.312·41-s + 0.914·43-s + 1.45·47-s + 1/7·49-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 - T \) | |
| good | 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 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 |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 12 T + p T^{2} \) | 1.61.m |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + 10 T + p T^{2} \) | 1.71.k |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−7.48919301543378268236009094915, −6.67001200400514998461131437340, −6.35832142593165731104405221796, −5.58369250511147231696408873723, −4.59553604464225747329684598584, −3.90687010515235259367548642118, −2.77624237097726961343072868023, −2.22915371909794712415784004590, −1.18405987245552524064844849436, 0,
1.18405987245552524064844849436, 2.22915371909794712415784004590, 2.77624237097726961343072868023, 3.90687010515235259367548642118, 4.59553604464225747329684598584, 5.58369250511147231696408873723, 6.35832142593165731104405221796, 6.67001200400514998461131437340, 7.48919301543378268236009094915
