| L(s) = 1 | − 5-s − 7-s − 6·11-s + 4·13-s + 3·17-s + 2·19-s + 2·23-s − 4·25-s + 6·29-s + 4·31-s + 35-s + 5·37-s + 5·41-s + 9·43-s − 5·47-s + 49-s − 6·53-s + 6·55-s + 7·59-s + 14·61-s − 4·65-s + 12·67-s − 8·71-s − 10·73-s + 6·77-s + 5·79-s − 11·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.377·7-s − 1.80·11-s + 1.10·13-s + 0.727·17-s + 0.458·19-s + 0.417·23-s − 4/5·25-s + 1.11·29-s + 0.718·31-s + 0.169·35-s + 0.821·37-s + 0.780·41-s + 1.37·43-s − 0.729·47-s + 1/7·49-s − 0.824·53-s + 0.809·55-s + 0.911·59-s + 1.79·61-s − 0.496·65-s + 1.46·67-s − 0.949·71-s − 1.17·73-s + 0.683·77-s + 0.562·79-s − 1.20·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.360037733\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.360037733\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 5 T + p T^{2} \) | 1.37.af |
| 41 | \( 1 - 5 T + p T^{2} \) | 1.41.af |
| 43 | \( 1 - 9 T + p T^{2} \) | 1.43.aj |
| 47 | \( 1 + 5 T + p T^{2} \) | 1.47.f |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 7 T + p T^{2} \) | 1.59.ah |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 5 T + p T^{2} \) | 1.79.af |
| 83 | \( 1 + 11 T + p T^{2} \) | 1.83.l |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−9.612192944412722556202150775350, −8.431238567626249734522622514695, −7.981459185083832488608522600255, −7.19741903344551563894052091570, −6.09797569521616858066133504609, −5.42530488196202943341547851940, −4.40257414407364560450760734636, −3.35076291605718888331997816423, −2.54105990574533197533918554272, −0.826107243154137004518883426677,
0.826107243154137004518883426677, 2.54105990574533197533918554272, 3.35076291605718888331997816423, 4.40257414407364560450760734636, 5.42530488196202943341547851940, 6.09797569521616858066133504609, 7.19741903344551563894052091570, 7.981459185083832488608522600255, 8.431238567626249734522622514695, 9.612192944412722556202150775350
