| L(s) = 1 | − 2·5-s − 4·11-s − 6·19-s − 2·23-s − 25-s + 6·29-s − 8·31-s − 2·37-s − 4·41-s + 2·43-s − 4·47-s − 7·49-s − 6·53-s + 8·55-s + 4·59-s − 2·61-s + 10·67-s − 4·71-s − 2·73-s − 8·79-s − 83-s − 14·89-s + 12·95-s − 14·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1.20·11-s − 1.37·19-s − 0.417·23-s − 1/5·25-s + 1.11·29-s − 1.43·31-s − 0.328·37-s − 0.624·41-s + 0.304·43-s − 0.583·47-s − 49-s − 0.824·53-s + 1.07·55-s + 0.520·59-s − 0.256·61-s + 1.22·67-s − 0.474·71-s − 0.234·73-s − 0.900·79-s − 0.109·83-s − 1.48·89-s + 1.23·95-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47808 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 83 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
| 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
−15.22717112814865, −14.58240178978171, −14.16088594537204, −13.47429768574656, −12.89755383471055, −12.59755553023324, −12.08262975258250, −11.30321861434355, −11.11174150411841, −10.40072208016014, −10.05168441394857, −9.349716759331434, −8.575114878960932, −8.224501564729129, −7.834739493111389, −7.153533672034730, −6.648117353063030, −5.961759732900415, −5.312281752220685, −4.737839021587585, −4.135505086250044, −3.575273895942146, −2.851334119949671, −2.205681521501282, −1.409076406411352, 0, 0,
1.409076406411352, 2.205681521501282, 2.851334119949671, 3.575273895942146, 4.135505086250044, 4.737839021587585, 5.312281752220685, 5.961759732900415, 6.648117353063030, 7.153533672034730, 7.834739493111389, 8.224501564729129, 8.575114878960932, 9.349716759331434, 10.05168441394857, 10.40072208016014, 11.11174150411841, 11.30321861434355, 12.08262975258250, 12.59755553023324, 12.89755383471055, 13.47429768574656, 14.16088594537204, 14.58240178978171, 15.22717112814865
