| L(s) = 1 | − 3-s − 4·5-s − 7-s + 9-s + 11-s − 6·13-s + 4·15-s − 2·19-s + 21-s − 4·23-s + 11·25-s − 27-s + 2·29-s + 2·31-s − 33-s + 4·35-s − 2·37-s + 6·39-s − 4·43-s − 4·45-s − 6·47-s + 49-s − 2·53-s − 4·55-s + 2·57-s − 14·61-s − 63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.78·5-s − 0.377·7-s + 1/3·9-s + 0.301·11-s − 1.66·13-s + 1.03·15-s − 0.458·19-s + 0.218·21-s − 0.834·23-s + 11/5·25-s − 0.192·27-s + 0.371·29-s + 0.359·31-s − 0.174·33-s + 0.676·35-s − 0.328·37-s + 0.960·39-s − 0.609·43-s − 0.596·45-s − 0.875·47-s + 1/7·49-s − 0.274·53-s − 0.539·55-s + 0.264·57-s − 1.79·61-s − 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 11 | \( 1 - T \) | |
| good | 5 | \( 1 + 4 T + p T^{2} \) | 1.5.e |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 2 T + p T^{2} \) | 1.83.ac |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−16.56967913721091, −16.16838666780415, −15.37775448765881, −15.23278754832496, −14.54078680487807, −13.97774302590271, −13.00407977775262, −12.54003209994791, −11.90412657230937, −11.86552419492383, −11.13711225477739, −10.36409598105143, −10.01168269872835, −9.148650788291690, −8.510967560979305, −7.754693036440342, −7.445331110384088, −6.763953717530142, −6.198872928089458, −5.209418401656287, −4.539742974076175, −4.198248714442164, −3.314929085696920, −2.655001923426188, −1.428275387260502, 0, 0,
1.428275387260502, 2.655001923426188, 3.314929085696920, 4.198248714442164, 4.539742974076175, 5.209418401656287, 6.198872928089458, 6.763953717530142, 7.445331110384088, 7.754693036440342, 8.510967560979305, 9.148650788291690, 10.01168269872835, 10.36409598105143, 11.13711225477739, 11.86552419492383, 11.90412657230937, 12.54003209994791, 13.00407977775262, 13.97774302590271, 14.54078680487807, 15.23278754832496, 15.37775448765881, 16.16838666780415, 16.56967913721091
