| L(s) = 1 | + 3-s − 2·4-s − 3·5-s − 4·7-s + 9-s − 2·12-s − 4·13-s − 3·15-s + 4·16-s + 3·17-s − 19-s + 6·20-s − 4·21-s + 6·23-s + 4·25-s + 27-s + 8·28-s − 6·29-s − 10·31-s + 12·35-s − 2·36-s − 7·37-s − 4·39-s + 2·43-s − 3·45-s − 3·47-s + 4·48-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s − 1.34·5-s − 1.51·7-s + 1/3·9-s − 0.577·12-s − 1.10·13-s − 0.774·15-s + 16-s + 0.727·17-s − 0.229·19-s + 1.34·20-s − 0.872·21-s + 1.25·23-s + 4/5·25-s + 0.192·27-s + 1.51·28-s − 1.11·29-s − 1.79·31-s + 2.02·35-s − 1/3·36-s − 1.15·37-s − 0.640·39-s + 0.304·43-s − 0.447·45-s − 0.437·47-s + 0.577·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 219 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 219 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) | |
| 73 | \( 1 - T \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 + 9 T + p T^{2} \) | 1.59.j |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 + 13 T + p T^{2} \) | 1.67.n |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 79 | \( 1 - 11 T + p T^{2} \) | 1.79.al |
| 83 | \( 1 - 15 T + p T^{2} \) | 1.83.ap |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 97 | \( 1 - 5 T + p T^{2} \) | 1.97.af |
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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
−12.23179896333458291697489196409, −10.69641796523503644619736991575, −9.564050761603220341273246198979, −9.017381962638777474294278722387, −7.78807751320390520795139874651, −7.04464507770050533616860795119, −5.28896520186268148672068535704, −3.92225433818331354547499519639, −3.21497761793707302576653608304, 0,
3.21497761793707302576653608304, 3.92225433818331354547499519639, 5.28896520186268148672068535704, 7.04464507770050533616860795119, 7.78807751320390520795139874651, 9.017381962638777474294278722387, 9.564050761603220341273246198979, 10.69641796523503644619736991575, 12.23179896333458291697489196409
