| L(s) = 1 | + 3-s − 5-s + 9-s − 11-s + 13-s − 15-s + 17-s − 7·19-s − 3·23-s − 4·25-s + 27-s + 3·29-s + 8·31-s − 33-s − 7·37-s + 39-s − 8·41-s + 7·43-s − 45-s + 8·47-s + 51-s + 10·53-s + 55-s − 7·57-s − 4·59-s + 7·61-s − 65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.301·11-s + 0.277·13-s − 0.258·15-s + 0.242·17-s − 1.60·19-s − 0.625·23-s − 4/5·25-s + 0.192·27-s + 0.557·29-s + 1.43·31-s − 0.174·33-s − 1.15·37-s + 0.160·39-s − 1.24·41-s + 1.06·43-s − 0.149·45-s + 1.16·47-s + 0.140·51-s + 1.37·53-s + 0.134·55-s − 0.927·57-s − 0.520·59-s + 0.896·61-s − 0.124·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 17 | \( 1 - T + p T^{2} \) | 1.17.ab |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 - 7 T + p T^{2} \) | 1.43.ah |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 - T + p T^{2} \) | 1.73.ab |
| 79 | \( 1 + 2 T + p T^{2} \) | 1.79.c |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−13.79106261813500, −13.42331889354712, −12.81274346370361, −12.33718929794766, −11.91692127701348, −11.48985333489881, −10.73608085286510, −10.26749066502712, −10.13106037259915, −9.313894570843606, −8.676827294461804, −8.481530172132636, −7.942748870071969, −7.485020206499109, −6.813238485362613, −6.404677880830984, −5.785000733785628, −5.187479507322650, −4.449261579847992, −4.043538339630648, −3.615240523689722, −2.803112139346572, −2.327691206200938, −1.712722279472753, −0.8316817360617364, 0,
0.8316817360617364, 1.712722279472753, 2.327691206200938, 2.803112139346572, 3.615240523689722, 4.043538339630648, 4.449261579847992, 5.187479507322650, 5.785000733785628, 6.404677880830984, 6.813238485362613, 7.485020206499109, 7.942748870071969, 8.481530172132636, 8.676827294461804, 9.313894570843606, 10.13106037259915, 10.26749066502712, 10.73608085286510, 11.48985333489881, 11.91692127701348, 12.33718929794766, 12.81274346370361, 13.42331889354712, 13.79106261813500
