| L(s) = 1 | − 3·3-s − 7-s + 6·9-s + 3·11-s − 13-s + 5·17-s − 8·19-s + 3·21-s − 2·23-s − 9·27-s − 29-s − 2·31-s − 9·33-s − 10·37-s + 3·39-s − 6·41-s + 4·43-s − 11·47-s + 49-s − 15·51-s − 6·53-s + 24·57-s − 10·59-s − 6·63-s + 10·67-s + 6·69-s + 10·73-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 0.377·7-s + 2·9-s + 0.904·11-s − 0.277·13-s + 1.21·17-s − 1.83·19-s + 0.654·21-s − 0.417·23-s − 1.73·27-s − 0.185·29-s − 0.359·31-s − 1.56·33-s − 1.64·37-s + 0.480·39-s − 0.937·41-s + 0.609·43-s − 1.60·47-s + 1/7·49-s − 2.10·51-s − 0.824·53-s + 3.17·57-s − 1.30·59-s − 0.755·63-s + 1.22·67-s + 0.722·69-s + 1.17·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| good | 3 | \( 1 + p T + p T^{2} \) | 1.3.d |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 - 5 T + p T^{2} \) | 1.17.af |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 11 T + p T^{2} \) | 1.47.l |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 7 T + p T^{2} \) | 1.79.h |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 8 T + p T^{2} \) | 1.89.ai |
| 97 | \( 1 + 3 T + p T^{2} \) | 1.97.d |
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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
−10.23289993314101448684647128494, −9.449814135625514329677059008330, −8.223881190328437078756964678651, −6.96649901971590149984227133098, −6.41433519070028327066970109988, −5.60873389286377035035591089929, −4.67130237358813591823357623580, −3.64130095908937728386793491367, −1.61644067753614716077188645392, 0,
1.61644067753614716077188645392, 3.64130095908937728386793491367, 4.67130237358813591823357623580, 5.60873389286377035035591089929, 6.41433519070028327066970109988, 6.96649901971590149984227133098, 8.223881190328437078756964678651, 9.449814135625514329677059008330, 10.23289993314101448684647128494
