| L(s) = 1 | + 5-s − 2·7-s − 3·9-s − 11-s + 6·17-s − 7·19-s − 23-s + 25-s + 7·29-s − 2·35-s − 2·37-s + 8·41-s − 43-s − 3·45-s + 47-s − 3·49-s − 4·53-s − 55-s − 6·59-s + 6·61-s + 6·63-s + 16·67-s − 11·71-s − 6·73-s + 2·77-s + 10·79-s + 9·81-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.755·7-s − 9-s − 0.301·11-s + 1.45·17-s − 1.60·19-s − 0.208·23-s + 1/5·25-s + 1.29·29-s − 0.338·35-s − 0.328·37-s + 1.24·41-s − 0.152·43-s − 0.447·45-s + 0.145·47-s − 3/7·49-s − 0.549·53-s − 0.134·55-s − 0.781·59-s + 0.768·61-s + 0.755·63-s + 1.95·67-s − 1.30·71-s − 0.702·73-s + 0.227·77-s + 1.12·79-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 148720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 148720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 11 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 29 | \( 1 - 7 T + p T^{2} \) | 1.29.ah |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 - T + p T^{2} \) | 1.47.ab |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 16 T + p T^{2} \) | 1.67.aq |
| 71 | \( 1 + 11 T + p T^{2} \) | 1.71.l |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 - 5 T + p T^{2} \) | 1.83.af |
| 89 | \( 1 + 9 T + p T^{2} \) | 1.89.j |
| 97 | \( 1 - T + p T^{2} \) | 1.97.ab |
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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
−13.65087962559966, −13.00775074668080, −12.64844755325368, −12.26001668165221, −11.77949602532025, −11.09920884467876, −10.66604710063624, −10.27694615499083, −9.692805958224846, −9.395086323960436, −8.659948301828146, −8.316973774627080, −7.885909495233276, −7.190634862154385, −6.534335475350587, −6.191616218348796, −5.774162398984497, −5.195895232121086, −4.653985425996162, −3.905513730246945, −3.371987535193725, −2.735766896270863, −2.427248067725601, −1.560251584948996, −0.7454479033361730, 0,
0.7454479033361730, 1.560251584948996, 2.427248067725601, 2.735766896270863, 3.371987535193725, 3.905513730246945, 4.653985425996162, 5.195895232121086, 5.774162398984497, 6.191616218348796, 6.534335475350587, 7.190634862154385, 7.885909495233276, 8.316973774627080, 8.659948301828146, 9.395086323960436, 9.692805958224846, 10.27694615499083, 10.66604710063624, 11.09920884467876, 11.77949602532025, 12.26001668165221, 12.64844755325368, 13.00775074668080, 13.65087962559966
