| L(s) = 1 | − 3-s − 2·4-s − 3·7-s + 9-s + 2·11-s + 2·12-s + 3·13-s + 4·16-s + 19-s + 3·21-s − 23-s − 27-s + 6·28-s + 31-s − 2·33-s − 2·36-s − 6·37-s − 3·39-s + 4·41-s + 5·43-s − 4·44-s − 10·47-s − 4·48-s + 2·49-s − 6·52-s − 4·53-s − 57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s − 1.13·7-s + 1/3·9-s + 0.603·11-s + 0.577·12-s + 0.832·13-s + 16-s + 0.229·19-s + 0.654·21-s − 0.208·23-s − 0.192·27-s + 1.13·28-s + 0.179·31-s − 0.348·33-s − 1/3·36-s − 0.986·37-s − 0.480·39-s + 0.624·41-s + 0.762·43-s − 0.603·44-s − 1.45·47-s − 0.577·48-s + 2/7·49-s − 0.832·52-s − 0.549·53-s − 0.132·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 23 | \( 1 + T \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 - 3 T + p T^{2} \) | 1.13.ad |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - T + p T^{2} \) | 1.31.ab |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 43 | \( 1 - 5 T + p T^{2} \) | 1.43.af |
| 47 | \( 1 + 10 T + p T^{2} \) | 1.47.k |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 + 7 T + p T^{2} \) | 1.67.h |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 + 7 T + p T^{2} \) | 1.97.h |
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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
−9.070923600860846590016948253811, −8.294724833972019336232188362245, −7.24625628736045120036518613143, −6.31568589834864421198558318677, −5.80737232480115415534956781209, −4.75927499384694056830484894059, −3.89876603085030278002719431366, −3.15525744730468461603496260385, −1.31675359847467968155467130609, 0,
1.31675359847467968155467130609, 3.15525744730468461603496260385, 3.89876603085030278002719431366, 4.75927499384694056830484894059, 5.80737232480115415534956781209, 6.31568589834864421198558318677, 7.24625628736045120036518613143, 8.294724833972019336232188362245, 9.070923600860846590016948253811
