| L(s) = 1 | − 3-s − 5-s + 9-s − 2·13-s + 15-s + 4·19-s − 8·23-s + 25-s − 27-s + 2·29-s − 4·31-s − 6·37-s + 2·39-s + 6·41-s − 8·43-s − 45-s − 12·47-s − 7·49-s + 6·53-s − 4·57-s + 12·59-s + 2·61-s + 2·65-s + 16·67-s + 8·69-s − 12·71-s + 6·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.554·13-s + 0.258·15-s + 0.917·19-s − 1.66·23-s + 1/5·25-s − 0.192·27-s + 0.371·29-s − 0.718·31-s − 0.986·37-s + 0.320·39-s + 0.937·41-s − 1.21·43-s − 0.149·45-s − 1.75·47-s − 49-s + 0.824·53-s − 0.529·57-s + 1.56·59-s + 0.256·61-s + 0.248·65-s + 1.95·67-s + 0.963·69-s − 1.42·71-s + 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 16 T + p T^{2} \) | 1.67.aq |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−14.39058134825614, −14.04251136815829, −13.18850131026712, −12.96773398779540, −12.22448759287634, −11.85840301720599, −11.52581175096386, −10.99227924817680, −10.26477590556744, −9.913734847470679, −9.529012558960037, −8.693410795818445, −8.187641281275514, −7.728398021995244, −7.093152146405523, −6.697338996030123, −6.024019747195615, −5.417847752932216, −4.974712105324044, −4.374091804027185, −3.647480006458336, −3.262449159045939, −2.255607461372881, −1.715245520589298, −0.7456441698276771, 0,
0.7456441698276771, 1.715245520589298, 2.255607461372881, 3.262449159045939, 3.647480006458336, 4.374091804027185, 4.974712105324044, 5.417847752932216, 6.024019747195615, 6.697338996030123, 7.093152146405523, 7.728398021995244, 8.187641281275514, 8.693410795818445, 9.529012558960037, 9.913734847470679, 10.26477590556744, 10.99227924817680, 11.52581175096386, 11.85840301720599, 12.22448759287634, 12.96773398779540, 13.18850131026712, 14.04251136815829, 14.39058134825614
