| L(s) = 1 | − 3·5-s − 2·7-s + 2·11-s + 3·17-s + 6·19-s + 6·23-s + 4·25-s − 5·29-s + 6·35-s + 37-s + 5·41-s + 6·43-s − 6·47-s − 3·49-s + 3·53-s − 6·55-s + 12·59-s + 7·61-s − 10·67-s + 10·71-s − 11·73-s − 4·77-s − 8·79-s − 14·83-s − 9·85-s + 10·89-s − 18·95-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 0.755·7-s + 0.603·11-s + 0.727·17-s + 1.37·19-s + 1.25·23-s + 4/5·25-s − 0.928·29-s + 1.01·35-s + 0.164·37-s + 0.780·41-s + 0.914·43-s − 0.875·47-s − 3/7·49-s + 0.412·53-s − 0.809·55-s + 1.56·59-s + 0.896·61-s − 1.22·67-s + 1.18·71-s − 1.28·73-s − 0.455·77-s − 0.900·79-s − 1.53·83-s − 0.976·85-s + 1.05·89-s − 1.84·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 97344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - T + p T^{2} \) | 1.37.ab |
| 41 | \( 1 - 5 T + p T^{2} \) | 1.41.af |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 + 10 T + p T^{2} \) | 1.67.k |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 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
−14.22148691301728, −13.24222911152605, −13.16247567452778, −12.50658463655540, −11.96338901206078, −11.65664145438319, −11.19330453122905, −10.74669545401838, −9.848567519340320, −9.705690287622674, −9.001786614488206, −8.632291113343302, −7.798404227971153, −7.591187428096719, −6.997095924156273, −6.637786609687340, −5.738854109381335, −5.420170096987437, −4.642193147949619, −4.029793788966496, −3.561246919621724, −3.136603881573360, −2.534342506899437, −1.360474436874871, −0.8528527434379321, 0,
0.8528527434379321, 1.360474436874871, 2.534342506899437, 3.136603881573360, 3.561246919621724, 4.029793788966496, 4.642193147949619, 5.420170096987437, 5.738854109381335, 6.637786609687340, 6.997095924156273, 7.591187428096719, 7.798404227971153, 8.632291113343302, 9.001786614488206, 9.705690287622674, 9.848567519340320, 10.74669545401838, 11.19330453122905, 11.65664145438319, 11.96338901206078, 12.50658463655540, 13.16247567452778, 13.24222911152605, 14.22148691301728
