| L(s) = 1 | + 3-s − 7-s + 9-s − 11-s + 3·13-s + 3·17-s + 5·19-s − 21-s − 4·23-s + 27-s + 8·29-s − 6·31-s − 33-s − 7·37-s + 3·39-s − 5·41-s + 8·43-s + 4·47-s + 49-s + 3·51-s − 53-s + 5·57-s − 4·59-s − 7·61-s − 63-s − 5·67-s − 4·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.301·11-s + 0.832·13-s + 0.727·17-s + 1.14·19-s − 0.218·21-s − 0.834·23-s + 0.192·27-s + 1.48·29-s − 1.07·31-s − 0.174·33-s − 1.15·37-s + 0.480·39-s − 0.780·41-s + 1.21·43-s + 0.583·47-s + 1/7·49-s + 0.420·51-s − 0.137·53-s + 0.662·57-s − 0.520·59-s − 0.896·61-s − 0.125·63-s − 0.610·67-s − 0.481·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.923256191\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.923256191\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 11 | \( 1 + T \) | |
| good | 13 | \( 1 - 3 T + p T^{2} \) | 1.13.ad |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + T + p T^{2} \) | 1.53.b |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 7 T + p T^{2} \) | 1.61.h |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 - 5 T + p T^{2} \) | 1.71.af |
| 73 | \( 1 - 9 T + p T^{2} \) | 1.73.aj |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 4 T + p T^{2} \) | 1.97.ae |
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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
−15.56281505420063, −14.98849388109411, −14.16812092157000, −13.85995374383891, −13.58405192030345, −12.67978095021549, −12.30066529252495, −11.82653624103189, −10.96822238013570, −10.49640592047643, −9.942963605460162, −9.377662157573169, −8.823565727866648, −8.231712896303563, −7.658366333096080, −7.157704277168652, −6.385720828749791, −5.802631244344258, −5.166668382646851, −4.409046457088256, −3.479863530603742, −3.325166052494829, −2.377891403813641, −1.546748609678371, −0.6872976415323615,
0.6872976415323615, 1.546748609678371, 2.377891403813641, 3.325166052494829, 3.479863530603742, 4.409046457088256, 5.166668382646851, 5.802631244344258, 6.385720828749791, 7.157704277168652, 7.658366333096080, 8.231712896303563, 8.823565727866648, 9.377662157573169, 9.942963605460162, 10.49640592047643, 10.96822238013570, 11.82653624103189, 12.30066529252495, 12.67978095021549, 13.58405192030345, 13.85995374383891, 14.16812092157000, 14.98849388109411, 15.56281505420063
