| L(s) = 1 | + 5-s − 2·7-s − 2·11-s − 13-s − 3·17-s + 2·19-s + 6·23-s − 4·25-s + 29-s + 8·31-s − 2·35-s − 37-s + 2·41-s + 10·43-s + 4·47-s − 3·49-s − 10·53-s − 2·55-s − 4·59-s − 9·61-s − 65-s − 14·67-s + 10·71-s − 9·73-s + 4·77-s − 10·79-s − 12·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.755·7-s − 0.603·11-s − 0.277·13-s − 0.727·17-s + 0.458·19-s + 1.25·23-s − 4/5·25-s + 0.185·29-s + 1.43·31-s − 0.338·35-s − 0.164·37-s + 0.312·41-s + 1.52·43-s + 0.583·47-s − 3/7·49-s − 1.37·53-s − 0.269·55-s − 0.520·59-s − 1.15·61-s − 0.124·65-s − 1.71·67-s + 1.18·71-s − 1.05·73-s + 0.455·77-s − 1.12·79-s − 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - T + p T^{2} \) | 1.29.ab |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + T + p T^{2} \) | 1.37.b |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 9 T + p T^{2} \) | 1.61.j |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 + 9 T + p T^{2} \) | 1.73.j |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 11 T + p T^{2} \) | 1.89.l |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−7.75235202489596116987221153909, −7.15503344949864472377679945145, −6.33003240335358684324706249445, −5.80962453823670423576455096186, −4.90309400915679139099700198064, −4.22496278157208158013903244704, −3.02321207443654244445473256446, −2.60761435707141134995359674587, −1.34610159131558151670679516246, 0,
1.34610159131558151670679516246, 2.60761435707141134995359674587, 3.02321207443654244445473256446, 4.22496278157208158013903244704, 4.90309400915679139099700198064, 5.80962453823670423576455096186, 6.33003240335358684324706249445, 7.15503344949864472377679945145, 7.75235202489596116987221153909
