| L(s) = 1 | − 2·3-s + 9-s + 11-s − 5·13-s − 7·19-s − 3·23-s + 4·27-s + 3·29-s + 5·31-s − 2·33-s − 4·37-s + 10·39-s − 12·41-s − 5·43-s + 6·53-s + 14·57-s + 12·59-s + 10·61-s − 14·67-s + 6·69-s − 3·71-s − 8·73-s + 4·79-s − 11·81-s − 15·83-s − 6·87-s − 3·89-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/3·9-s + 0.301·11-s − 1.38·13-s − 1.60·19-s − 0.625·23-s + 0.769·27-s + 0.557·29-s + 0.898·31-s − 0.348·33-s − 0.657·37-s + 1.60·39-s − 1.87·41-s − 0.762·43-s + 0.824·53-s + 1.85·57-s + 1.56·59-s + 1.28·61-s − 1.71·67-s + 0.722·69-s − 0.356·71-s − 0.936·73-s + 0.450·79-s − 1.22·81-s − 1.64·83-s − 0.643·87-s − 0.317·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 215600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 215600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 + 3 T + p T^{2} \) | 1.71.d |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 15 T + p T^{2} \) | 1.83.p |
| 89 | \( 1 + 3 T + p T^{2} \) | 1.89.d |
| 97 | \( 1 - 13 T + p T^{2} \) | 1.97.an |
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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
−13.20980667837503, −12.59650126653366, −12.06211858159244, −11.89964115834824, −11.56590532573759, −10.85891217707022, −10.38925906162455, −10.05033327307609, −9.794978782101738, −8.824378123315609, −8.513225527459724, −8.166639062962214, −7.238612486249526, −6.949958723939219, −6.516345722706888, −6.009225095583310, −5.489466531311912, −4.976755715970547, −4.548066103579614, −4.103661210087136, −3.325752612105241, −2.642723083687284, −2.102508200702464, −1.447505656264711, −0.5280766810687592, 0,
0.5280766810687592, 1.447505656264711, 2.102508200702464, 2.642723083687284, 3.325752612105241, 4.103661210087136, 4.548066103579614, 4.976755715970547, 5.489466531311912, 6.009225095583310, 6.516345722706888, 6.949958723939219, 7.238612486249526, 8.166639062962214, 8.513225527459724, 8.824378123315609, 9.794978782101738, 10.05033327307609, 10.38925906162455, 10.85891217707022, 11.56590532573759, 11.89964115834824, 12.06211858159244, 12.59650126653366, 13.20980667837503
