| L(s) = 1 | − 3·5-s + 4·7-s − 13-s − 3·17-s + 4·19-s + 4·25-s + 9·29-s + 4·31-s − 12·35-s − 37-s + 6·41-s − 8·43-s + 12·47-s + 9·49-s − 6·53-s − 61-s + 3·65-s + 4·67-s + 12·71-s + 11·73-s + 16·79-s + 12·83-s + 9·85-s − 3·89-s − 4·91-s − 12·95-s + 2·97-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 1.51·7-s − 0.277·13-s − 0.727·17-s + 0.917·19-s + 4/5·25-s + 1.67·29-s + 0.718·31-s − 2.02·35-s − 0.164·37-s + 0.937·41-s − 1.21·43-s + 1.75·47-s + 9/7·49-s − 0.824·53-s − 0.128·61-s + 0.372·65-s + 0.488·67-s + 1.42·71-s + 1.28·73-s + 1.80·79-s + 1.31·83-s + 0.976·85-s − 0.317·89-s − 0.419·91-s − 1.23·95-s + 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.504095745\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.504095745\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 3 | \( 1 \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + T + p T^{2} \) | 1.37.b |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 3 T + p T^{2} \) | 1.89.d |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−9.598605566629630565849257506423, −8.542414927665885183690210552564, −8.047221009677602422962699873621, −7.44571478799237129140639120122, −6.50974172394367970294419909175, −5.10609983558079824652464594116, −4.60529471992634973001671480527, −3.68824329939728705086964001391, −2.41615112845105918353887046447, −0.935395355311356120178486862505,
0.935395355311356120178486862505, 2.41615112845105918353887046447, 3.68824329939728705086964001391, 4.60529471992634973001671480527, 5.10609983558079824652464594116, 6.50974172394367970294419909175, 7.44571478799237129140639120122, 8.047221009677602422962699873621, 8.542414927665885183690210552564, 9.598605566629630565849257506423
