| L(s) = 1 | − 5-s − 4·11-s + 6·13-s + 4·17-s + 6·19-s + 25-s − 6·29-s + 4·31-s − 8·37-s + 10·41-s + 2·43-s + 10·47-s + 14·53-s + 4·55-s + 4·59-s − 8·61-s − 6·65-s − 6·67-s + 2·71-s + 10·73-s + 16·79-s + 8·83-s − 4·85-s + 2·89-s − 6·95-s − 2·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.20·11-s + 1.66·13-s + 0.970·17-s + 1.37·19-s + 1/5·25-s − 1.11·29-s + 0.718·31-s − 1.31·37-s + 1.56·41-s + 0.304·43-s + 1.45·47-s + 1.92·53-s + 0.539·55-s + 0.520·59-s − 1.02·61-s − 0.744·65-s − 0.733·67-s + 0.237·71-s + 1.17·73-s + 1.80·79-s + 0.878·83-s − 0.433·85-s + 0.211·89-s − 0.615·95-s − 0.203·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.217733827\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.217733827\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| good | 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 53 | \( 1 - 14 T + p T^{2} \) | 1.53.ao |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 + 6 T + p T^{2} \) | 1.67.g |
| 71 | \( 1 - 2 T + p T^{2} \) | 1.71.ac |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 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
−13.54124874382154, −12.89976808416405, −12.45169104863781, −11.96186623361256, −11.50176793134300, −10.93508707839596, −10.60322001795988, −10.14952850643731, −9.533667299495681, −8.925829434263418, −8.607076508880316, −7.921478616899471, −7.458401920608854, −7.356462563323254, −6.359355714950798, −5.889847036546676, −5.427005801915904, −5.025450439936512, −4.201152306595846, −3.586487610656467, −3.348940871119329, −2.587400509171252, −1.922190615882605, −0.9541935256283427, −0.6842034542009715,
0.6842034542009715, 0.9541935256283427, 1.922190615882605, 2.587400509171252, 3.348940871119329, 3.586487610656467, 4.201152306595846, 5.025450439936512, 5.427005801915904, 5.889847036546676, 6.359355714950798, 7.356462563323254, 7.458401920608854, 7.921478616899471, 8.607076508880316, 8.925829434263418, 9.533667299495681, 10.14952850643731, 10.60322001795988, 10.93508707839596, 11.50176793134300, 11.96186623361256, 12.45169104863781, 12.89976808416405, 13.54124874382154
