| L(s) = 1 | + 3-s − 5-s + 2·7-s + 9-s + 3·11-s − 4·13-s − 15-s − 5·19-s + 2·21-s − 4·23-s + 25-s + 27-s + 9·29-s + 3·33-s − 2·35-s + 6·37-s − 4·39-s + 5·41-s − 2·43-s − 45-s − 10·47-s − 3·49-s − 2·53-s − 3·55-s − 5·57-s − 7·59-s + 9·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s + 0.904·11-s − 1.10·13-s − 0.258·15-s − 1.14·19-s + 0.436·21-s − 0.834·23-s + 1/5·25-s + 0.192·27-s + 1.67·29-s + 0.522·33-s − 0.338·35-s + 0.986·37-s − 0.640·39-s + 0.780·41-s − 0.304·43-s − 0.149·45-s − 1.45·47-s − 3/7·49-s − 0.274·53-s − 0.404·55-s − 0.662·57-s − 0.911·59-s + 1.15·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 34680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 34680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 5 | \( 1 + T \) | |
| 17 | \( 1 \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 5 T + p T^{2} \) | 1.41.af |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 + 10 T + p T^{2} \) | 1.47.k |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 7 T + p T^{2} \) | 1.59.h |
| 61 | \( 1 - 9 T + p T^{2} \) | 1.61.aj |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + 5 T + p T^{2} \) | 1.71.f |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 3 T + p T^{2} \) | 1.79.d |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - T + p T^{2} \) | 1.89.ab |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.93086130766029, −14.72138869599469, −14.31813106778771, −13.85856244008422, −13.03829782618918, −12.61363708154056, −11.99593529395599, −11.64441102367687, −11.04980788156674, −10.36567233030288, −9.872624476426862, −9.313196191983663, −8.685381765183468, −8.052161906200756, −7.937668699241755, −7.040006137199564, −6.584303972492479, −5.962138180606221, −4.996744405395888, −4.433093085293741, −4.180987921897275, −3.241517163814125, −2.568898207843407, −1.893369268057414, −1.133693012149487, 0,
1.133693012149487, 1.893369268057414, 2.568898207843407, 3.241517163814125, 4.180987921897275, 4.433093085293741, 4.996744405395888, 5.962138180606221, 6.584303972492479, 7.040006137199564, 7.937668699241755, 8.052161906200756, 8.685381765183468, 9.313196191983663, 9.872624476426862, 10.36567233030288, 11.04980788156674, 11.64441102367687, 11.99593529395599, 12.61363708154056, 13.03829782618918, 13.85856244008422, 14.31813106778771, 14.72138869599469, 14.93086130766029
