| L(s) = 1 | − 3-s + 9-s − 4·11-s − 2·13-s − 2·17-s + 8·19-s − 4·23-s − 27-s + 6·29-s + 4·33-s + 2·37-s + 2·39-s + 6·41-s + 4·43-s − 12·47-s + 2·51-s − 6·53-s − 8·57-s + 12·59-s + 14·61-s − 12·67-s + 4·69-s + 2·73-s − 8·79-s + 81-s + 4·83-s − 6·87-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 1.20·11-s − 0.554·13-s − 0.485·17-s + 1.83·19-s − 0.834·23-s − 0.192·27-s + 1.11·29-s + 0.696·33-s + 0.328·37-s + 0.320·39-s + 0.937·41-s + 0.609·43-s − 1.75·47-s + 0.280·51-s − 0.824·53-s − 1.05·57-s + 1.56·59-s + 1.79·61-s − 1.46·67-s + 0.481·69-s + 0.234·73-s − 0.900·79-s + 1/9·81-s + 0.439·83-s − 0.643·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−13.13523522522944, −12.65595181529905, −12.13495146171537, −11.80171371645382, −11.25823502596950, −10.93835207958864, −10.24774386279392, −9.951925151941796, −9.594160972502410, −9.024573496160313, −8.241717260155738, −7.939780658441606, −7.546598914828864, −6.843163777933498, −6.621682251008898, −5.757396924040769, −5.459903825918103, −5.062247149089903, −4.438668844115439, −4.013775228239300, −3.109888332390381, −2.780854565412184, −2.160437036581740, −1.379425942360483, −0.6988975233762342, 0,
0.6988975233762342, 1.379425942360483, 2.160437036581740, 2.780854565412184, 3.109888332390381, 4.013775228239300, 4.438668844115439, 5.062247149089903, 5.459903825918103, 5.757396924040769, 6.621682251008898, 6.843163777933498, 7.546598914828864, 7.939780658441606, 8.241717260155738, 9.024573496160313, 9.594160972502410, 9.951925151941796, 10.24774386279392, 10.93835207958864, 11.25823502596950, 11.80171371645382, 12.13495146171537, 12.65595181529905, 13.13523522522944
