| L(s) = 1 | − 2·3-s + 5-s + 9-s − 11-s − 2·13-s − 2·15-s − 2·19-s − 4·23-s + 25-s + 4·27-s + 2·29-s + 8·31-s + 2·33-s + 10·37-s + 4·39-s + 4·43-s + 45-s + 8·47-s + 6·53-s − 55-s + 4·57-s − 6·59-s + 2·61-s − 2·65-s − 4·67-s + 8·69-s + 4·73-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.447·5-s + 1/3·9-s − 0.301·11-s − 0.554·13-s − 0.516·15-s − 0.458·19-s − 0.834·23-s + 1/5·25-s + 0.769·27-s + 0.371·29-s + 1.43·31-s + 0.348·33-s + 1.64·37-s + 0.640·39-s + 0.609·43-s + 0.149·45-s + 1.16·47-s + 0.824·53-s − 0.134·55-s + 0.529·57-s − 0.781·59-s + 0.256·61-s − 0.248·65-s − 0.488·67-s + 0.963·69-s + 0.468·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 86240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.565543204\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.565543204\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.02670833946742, −13.30002887208011, −12.88901575710804, −12.27880889707227, −11.96042760365849, −11.51626701200888, −10.90738873467040, −10.41678775458071, −10.14577189902810, −9.514346543538936, −8.973816487271336, −8.319155002494464, −7.797402089079600, −7.248926206622457, −6.513528619731993, −6.158905204102343, −5.773294494570597, −5.148343605261338, −4.567468820809185, −4.230603014696795, −3.236986387630693, −2.517806995372923, −2.100060978833947, −0.9893508445708529, −0.5240549150018642,
0.5240549150018642, 0.9893508445708529, 2.100060978833947, 2.517806995372923, 3.236986387630693, 4.230603014696795, 4.567468820809185, 5.148343605261338, 5.773294494570597, 6.158905204102343, 6.513528619731993, 7.248926206622457, 7.797402089079600, 8.319155002494464, 8.973816487271336, 9.514346543538936, 10.14577189902810, 10.41678775458071, 10.90738873467040, 11.51626701200888, 11.96042760365849, 12.27880889707227, 12.88901575710804, 13.30002887208011, 14.02670833946742
