| L(s) = 1 | + 3-s − 2·5-s − 2·7-s + 9-s + 4·11-s − 2·15-s + 6·17-s − 2·21-s − 4·23-s − 25-s + 27-s + 8·31-s + 4·33-s + 4·35-s − 2·37-s − 41-s + 4·43-s − 2·45-s − 2·47-s − 3·49-s + 6·51-s − 4·53-s − 8·55-s − 12·59-s + 6·61-s − 2·63-s − 8·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s − 0.755·7-s + 1/3·9-s + 1.20·11-s − 0.516·15-s + 1.45·17-s − 0.436·21-s − 0.834·23-s − 1/5·25-s + 0.192·27-s + 1.43·31-s + 0.696·33-s + 0.676·35-s − 0.328·37-s − 0.156·41-s + 0.609·43-s − 0.298·45-s − 0.291·47-s − 3/7·49-s + 0.840·51-s − 0.549·53-s − 1.07·55-s − 1.56·59-s + 0.768·61-s − 0.251·63-s − 0.977·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.030165249\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.030165249\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 41 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
| 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
−7.969727308294545603448143891736, −7.25576929161445131422114725365, −6.48678018198950831330847799717, −5.97827649624082764974761379566, −4.87110217160159863781362581744, −4.04169142747632568645088344671, −3.54508009351102867585197149476, −2.94638528742611753510219401400, −1.74522566256138329988285267016, −0.70195388854807766803584326900,
0.70195388854807766803584326900, 1.74522566256138329988285267016, 2.94638528742611753510219401400, 3.54508009351102867585197149476, 4.04169142747632568645088344671, 4.87110217160159863781362581744, 5.97827649624082764974761379566, 6.48678018198950831330847799717, 7.25576929161445131422114725365, 7.969727308294545603448143891736
