| L(s) = 1 | + 2·3-s + 2·5-s + 4·7-s + 9-s − 2·11-s − 2·13-s + 4·15-s + 2·17-s − 2·19-s + 8·21-s + 4·23-s − 25-s − 4·27-s + 6·29-s − 4·33-s + 8·35-s − 10·37-s − 4·39-s − 6·41-s + 6·43-s + 2·45-s + 8·47-s + 9·49-s + 4·51-s + 6·53-s − 4·55-s − 4·57-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.894·5-s + 1.51·7-s + 1/3·9-s − 0.603·11-s − 0.554·13-s + 1.03·15-s + 0.485·17-s − 0.458·19-s + 1.74·21-s + 0.834·23-s − 1/5·25-s − 0.769·27-s + 1.11·29-s − 0.696·33-s + 1.35·35-s − 1.64·37-s − 0.640·39-s − 0.937·41-s + 0.914·43-s + 0.298·45-s + 1.16·47-s + 9/7·49-s + 0.560·51-s + 0.824·53-s − 0.539·55-s − 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 31 | \( 1 \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 14 T + p T^{2} \) | 1.59.o |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 10 T + p T^{2} \) | 1.67.k |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 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
−13.75255720490269, −13.60798447333070, −12.99991617514051, −12.28179020625680, −11.96803463036860, −11.37515039126253, −10.64761329400192, −10.38400634154552, −9.978201606479185, −9.103505449259925, −8.944850728491064, −8.422121112916857, −7.978613729966330, −7.366559050969736, −7.173103200722483, −6.200760668325727, −5.589341665796436, −5.287092266463632, −4.546717909053271, −4.244362864125461, −3.207346198552329, −2.839430235599612, −2.251333330598688, −1.701988004885409, −1.251001137846504, 0,
1.251001137846504, 1.701988004885409, 2.251333330598688, 2.839430235599612, 3.207346198552329, 4.244362864125461, 4.546717909053271, 5.287092266463632, 5.589341665796436, 6.200760668325727, 7.173103200722483, 7.366559050969736, 7.978613729966330, 8.422121112916857, 8.944850728491064, 9.103505449259925, 9.978201606479185, 10.38400634154552, 10.64761329400192, 11.37515039126253, 11.96803463036860, 12.28179020625680, 12.99991617514051, 13.60798447333070, 13.75255720490269
