| L(s) = 1 | + 4·7-s − 2·11-s − 4·13-s + 17-s − 5·19-s + 5·23-s − 8·29-s + 7·31-s + 6·37-s − 6·41-s + 2·43-s + 8·47-s + 9·49-s + 9·53-s − 4·59-s + 13·61-s + 10·67-s + 6·71-s + 6·73-s − 8·77-s + 9·79-s − 17·83-s + 6·89-s − 16·91-s + 8·97-s − 12·101-s − 4·103-s + ⋯ |
| L(s) = 1 | + 1.51·7-s − 0.603·11-s − 1.10·13-s + 0.242·17-s − 1.14·19-s + 1.04·23-s − 1.48·29-s + 1.25·31-s + 0.986·37-s − 0.937·41-s + 0.304·43-s + 1.16·47-s + 9/7·49-s + 1.23·53-s − 0.520·59-s + 1.66·61-s + 1.22·67-s + 0.712·71-s + 0.702·73-s − 0.911·77-s + 1.01·79-s − 1.86·83-s + 0.635·89-s − 1.67·91-s + 0.812·97-s − 1.19·101-s − 0.394·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.115445908\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.115445908\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| good | 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - T + p T^{2} \) | 1.17.ab |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 - 5 T + p T^{2} \) | 1.23.af |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 - 7 T + p T^{2} \) | 1.31.ah |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 13 T + p T^{2} \) | 1.61.an |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 9 T + p T^{2} \) | 1.79.aj |
| 83 | \( 1 + 17 T + p T^{2} \) | 1.83.r |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−8.200236871894429009128646007068, −7.48712254438609359127251495498, −6.96362131242169607346932744687, −5.86978245848317218524525630182, −5.12238460126295364130070310602, −4.68463194282086038135912565474, −3.83377998861779427686077657642, −2.55851900914552522563054013540, −2.03611294694083801808420297238, −0.78206232359650740359884138807,
0.78206232359650740359884138807, 2.03611294694083801808420297238, 2.55851900914552522563054013540, 3.83377998861779427686077657642, 4.68463194282086038135912565474, 5.12238460126295364130070310602, 5.86978245848317218524525630182, 6.96362131242169607346932744687, 7.48712254438609359127251495498, 8.200236871894429009128646007068
