| L(s) = 1 | − 3-s − 5-s − 7-s + 9-s − 13-s + 15-s + 2·17-s − 4·19-s + 21-s − 8·23-s + 25-s − 27-s − 6·29-s − 4·31-s + 35-s − 6·37-s + 39-s − 10·41-s − 45-s − 12·47-s + 49-s − 2·51-s − 2·53-s + 4·57-s − 4·59-s + 10·61-s − 63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.277·13-s + 0.258·15-s + 0.485·17-s − 0.917·19-s + 0.218·21-s − 1.66·23-s + 1/5·25-s − 0.192·27-s − 1.11·29-s − 0.718·31-s + 0.169·35-s − 0.986·37-s + 0.160·39-s − 1.56·41-s − 0.149·45-s − 1.75·47-s + 1/7·49-s − 0.280·51-s − 0.274·53-s + 0.529·57-s − 0.520·59-s + 1.28·61-s − 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2683960384\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2683960384\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 + T \) | |
| good | 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 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
−15.64640328098525, −14.97144951345180, −14.62694240652922, −13.89673490180905, −13.28310016229719, −12.70878435468508, −12.29710133540323, −11.67841306361235, −11.32661951052896, −10.47860810270428, −10.16710384414741, −9.571339908518117, −8.828014736922195, −8.222055151351723, −7.632850687007255, −7.031202297842573, −6.395483588852667, −5.870654560275758, −5.163718386740866, −4.562897363476855, −3.711717150108974, −3.390189671096053, −2.175932479319451, −1.584597541921457, −0.2111367719098465,
0.2111367719098465, 1.584597541921457, 2.175932479319451, 3.390189671096053, 3.711717150108974, 4.562897363476855, 5.163718386740866, 5.870654560275758, 6.395483588852667, 7.031202297842573, 7.632850687007255, 8.222055151351723, 8.828014736922195, 9.571339908518117, 10.16710384414741, 10.47860810270428, 11.32661951052896, 11.67841306361235, 12.29710133540323, 12.70878435468508, 13.28310016229719, 13.89673490180905, 14.62694240652922, 14.97144951345180, 15.64640328098525
