| L(s) = 1 | − 5-s + 7-s − 3·13-s + 17-s + 5·19-s − 4·23-s + 25-s − 9·29-s + 6·31-s − 35-s + 11·37-s + 2·41-s − 2·43-s + 6·47-s − 6·49-s + 4·53-s + 4·59-s − 4·61-s + 3·65-s + 6·67-s − 5·71-s + 6·73-s − 14·79-s + 7·83-s − 85-s − 4·89-s − 3·91-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.377·7-s − 0.832·13-s + 0.242·17-s + 1.14·19-s − 0.834·23-s + 1/5·25-s − 1.67·29-s + 1.07·31-s − 0.169·35-s + 1.80·37-s + 0.312·41-s − 0.304·43-s + 0.875·47-s − 6/7·49-s + 0.549·53-s + 0.520·59-s − 0.512·61-s + 0.372·65-s + 0.733·67-s − 0.593·71-s + 0.702·73-s − 1.57·79-s + 0.768·83-s − 0.108·85-s − 0.423·89-s − 0.314·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 348480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 348480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| good | 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 13 | \( 1 + 3 T + p T^{2} \) | 1.13.d |
| 17 | \( 1 - T + p T^{2} \) | 1.17.ab |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 9 T + p T^{2} \) | 1.29.j |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 - 11 T + p T^{2} \) | 1.37.al |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 - 6 T + p T^{2} \) | 1.67.ag |
| 71 | \( 1 + 5 T + p T^{2} \) | 1.71.f |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 - 7 T + p T^{2} \) | 1.83.ah |
| 89 | \( 1 + 4 T + p T^{2} \) | 1.89.e |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−12.69076209001392, −12.29056501441372, −11.73791939884906, −11.58002314227815, −11.03355299515246, −10.59974332677945, −9.834811366290884, −9.690845186666791, −9.316454891529427, −8.493773731549141, −8.198385823470341, −7.628717247608146, −7.397031049063603, −6.921918190585672, −6.155934481122476, −5.790172621662782, −5.239484151842946, −4.790203523247476, −4.168248643392912, −3.867569639551033, −3.110177460407463, −2.652036728202105, −2.080051261935058, −1.361988091840553, −0.7562517609063039, 0,
0.7562517609063039, 1.361988091840553, 2.080051261935058, 2.652036728202105, 3.110177460407463, 3.867569639551033, 4.168248643392912, 4.790203523247476, 5.239484151842946, 5.790172621662782, 6.155934481122476, 6.921918190585672, 7.397031049063603, 7.628717247608146, 8.198385823470341, 8.493773731549141, 9.316454891529427, 9.690845186666791, 9.834811366290884, 10.59974332677945, 11.03355299515246, 11.58002314227815, 11.73791939884906, 12.29056501441372, 12.69076209001392
