| L(s) = 1 | − 5-s + 3·7-s + 6·13-s − 7·17-s + 5·19-s − 6·23-s + 25-s + 5·29-s + 3·31-s − 3·35-s + 3·37-s + 2·41-s + 4·43-s − 2·47-s + 2·49-s + 53-s − 10·59-s − 7·61-s − 6·65-s − 8·67-s + 7·71-s − 14·73-s + 10·79-s + 6·83-s + 7·85-s + 15·89-s + 18·91-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.13·7-s + 1.66·13-s − 1.69·17-s + 1.14·19-s − 1.25·23-s + 1/5·25-s + 0.928·29-s + 0.538·31-s − 0.507·35-s + 0.493·37-s + 0.312·41-s + 0.609·43-s − 0.291·47-s + 2/7·49-s + 0.137·53-s − 1.30·59-s − 0.896·61-s − 0.744·65-s − 0.977·67-s + 0.830·71-s − 1.63·73-s + 1.12·79-s + 0.658·83-s + 0.759·85-s + 1.58·89-s + 1.88·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.819258621\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.819258621\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 5 T + p T^{2} \) | 1.29.af |
| 31 | \( 1 - 3 T + p T^{2} \) | 1.31.ad |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - T + p T^{2} \) | 1.53.ab |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 + 7 T + p T^{2} \) | 1.61.h |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 7 T + p T^{2} \) | 1.71.ah |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 15 T + p T^{2} \) | 1.89.ap |
| 97 | \( 1 + 12 T + p T^{2} \) | 1.97.m |
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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
−13.87577429986575, −13.55754268576517, −13.03539148727083, −12.24467001901636, −11.87194598244317, −11.39197605407133, −10.97428099057313, −10.65461356121615, −9.988740476884894, −9.206936527915261, −8.849142467165065, −8.329061870494636, −7.855813129662641, −7.520430335589646, −6.634456837797447, −6.239842222820643, −5.746432760836162, −4.902139041484009, −4.487085576471120, −4.052454421801746, −3.351676537295797, −2.672436520528560, −1.871400863847277, −1.323615551704343, −0.5685469079744815,
0.5685469079744815, 1.323615551704343, 1.871400863847277, 2.672436520528560, 3.351676537295797, 4.052454421801746, 4.487085576471120, 4.902139041484009, 5.746432760836162, 6.239842222820643, 6.634456837797447, 7.520430335589646, 7.855813129662641, 8.329061870494636, 8.849142467165065, 9.206936527915261, 9.988740476884894, 10.65461356121615, 10.97428099057313, 11.39197605407133, 11.87194598244317, 12.24467001901636, 13.03539148727083, 13.55754268576517, 13.87577429986575
