| L(s) = 1 | + 2·2-s + 3-s + 2·4-s − 5-s + 2·6-s + 5·7-s + 9-s − 2·10-s + 2·11-s + 2·12-s + 10·14-s − 15-s − 4·16-s + 2·17-s + 2·18-s − 2·20-s + 5·21-s + 4·22-s + 6·23-s + 25-s + 27-s + 10·28-s − 4·29-s − 2·30-s − 7·31-s − 8·32-s + 2·33-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 4-s − 0.447·5-s + 0.816·6-s + 1.88·7-s + 1/3·9-s − 0.632·10-s + 0.603·11-s + 0.577·12-s + 2.67·14-s − 0.258·15-s − 16-s + 0.485·17-s + 0.471·18-s − 0.447·20-s + 1.09·21-s + 0.852·22-s + 1.25·23-s + 1/5·25-s + 0.192·27-s + 1.88·28-s − 0.742·29-s − 0.365·30-s − 1.25·31-s − 1.41·32-s + 0.348·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.501285236\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.501285236\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 2 | \( 1 - p T + p T^{2} \) | 1.2.ac |
| 7 | \( 1 - 5 T + p T^{2} \) | 1.7.af |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + 7 T + p T^{2} \) | 1.31.h |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 13 T + p T^{2} \) | 1.61.n |
| 67 | \( 1 + 7 T + p T^{2} \) | 1.67.h |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 15 T + p T^{2} \) | 1.73.ap |
| 79 | \( 1 - 3 T + p T^{2} \) | 1.79.ad |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 + 5 T + p T^{2} \) | 1.97.f |
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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
−8.847039756700737460746614498221, −7.962892854706047727128101299377, −7.40761087873862071665807911297, −6.51559411539978021603868071441, −5.33093616174838671754167090927, −4.97085438122322756882993905237, −4.07848782734203899777308010471, −3.50912108083906362485072638882, −2.37594774574149352432378533976, −1.37536554166182890266287628361,
1.37536554166182890266287628361, 2.37594774574149352432378533976, 3.50912108083906362485072638882, 4.07848782734203899777308010471, 4.97085438122322756882993905237, 5.33093616174838671754167090927, 6.51559411539978021603868071441, 7.40761087873862071665807911297, 7.962892854706047727128101299377, 8.847039756700737460746614498221
