| L(s) = 1 | − 2·5-s − 4·7-s + 4·13-s − 6·17-s − 4·19-s − 2·23-s − 25-s + 4·31-s + 8·35-s − 2·37-s + 2·41-s − 8·43-s + 9·49-s + 6·53-s + 4·59-s − 10·61-s − 8·65-s − 4·67-s + 12·71-s + 2·73-s + 10·79-s + 83-s + 12·85-s + 2·89-s − 16·91-s + 8·95-s + 6·97-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1.51·7-s + 1.10·13-s − 1.45·17-s − 0.917·19-s − 0.417·23-s − 1/5·25-s + 0.718·31-s + 1.35·35-s − 0.328·37-s + 0.312·41-s − 1.21·43-s + 9/7·49-s + 0.824·53-s + 0.520·59-s − 1.28·61-s − 0.992·65-s − 0.488·67-s + 1.42·71-s + 0.234·73-s + 1.12·79-s + 0.109·83-s + 1.30·85-s + 0.211·89-s − 1.67·91-s + 0.820·95-s + 0.609·97-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 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−15.11844296510510, −14.26756151421920, −13.51677017820208, −13.38638192952116, −12.83767552035564, −12.24989067640723, −11.80731919279491, −11.17717276758411, −10.74821314854606, −10.19157234210520, −9.619909509825796, −8.971202459249564, −8.576004377729814, −8.080757388701969, −7.345445145299373, −6.693293256397119, −6.353654762861307, −5.951082913739422, −4.958633234047375, −4.283058840266103, −3.774119796523465, −3.374246391642660, −2.560183061685185, −1.860158875968290, −0.6710982969914153, 0,
0.6710982969914153, 1.860158875968290, 2.560183061685185, 3.374246391642660, 3.774119796523465, 4.283058840266103, 4.958633234047375, 5.951082913739422, 6.353654762861307, 6.693293256397119, 7.345445145299373, 8.080757388701969, 8.576004377729814, 8.971202459249564, 9.619909509825796, 10.19157234210520, 10.74821314854606, 11.17717276758411, 11.80731919279491, 12.24989067640723, 12.83767552035564, 13.38638192952116, 13.51677017820208, 14.26756151421920, 15.11844296510510
