| L(s) = 1 | − 2-s + 4-s − 5-s + 7-s − 8-s + 10-s − 4·11-s + 2·13-s − 14-s + 16-s − 8·19-s − 20-s + 4·22-s + 25-s − 2·26-s + 28-s − 2·29-s + 8·31-s − 32-s − 35-s + 2·37-s + 8·38-s + 40-s + 2·41-s + 4·43-s − 4·44-s + 8·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.377·7-s − 0.353·8-s + 0.316·10-s − 1.20·11-s + 0.554·13-s − 0.267·14-s + 1/4·16-s − 1.83·19-s − 0.223·20-s + 0.852·22-s + 1/5·25-s − 0.392·26-s + 0.188·28-s − 0.371·29-s + 1.43·31-s − 0.176·32-s − 0.169·35-s + 0.328·37-s + 1.29·38-s + 0.158·40-s + 0.312·41-s + 0.609·43-s − 0.603·44-s + 1.16·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 182070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 182070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 - T \) | |
| 17 | \( 1 \) | |
| good | 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−13.25952775686100, −12.92751461089109, −12.29282107876363, −12.07324394804999, −11.19945735017312, −10.97456241420307, −10.70476876202052, −10.04777212558107, −9.725752499887359, −8.906878647200330, −8.529267309438003, −8.231987264624627, −7.746857455495594, −7.253393710708614, −6.700127992979045, −6.123835790520487, −5.697916655410158, −5.050339256454816, −4.346780385236036, −4.084324520864932, −3.250703957320343, −2.506950143248016, −2.307587388147208, −1.407821174091147, −0.7031728802117200, 0,
0.7031728802117200, 1.407821174091147, 2.307587388147208, 2.506950143248016, 3.250703957320343, 4.084324520864932, 4.346780385236036, 5.050339256454816, 5.697916655410158, 6.123835790520487, 6.700127992979045, 7.253393710708614, 7.746857455495594, 8.231987264624627, 8.529267309438003, 8.906878647200330, 9.725752499887359, 10.04777212558107, 10.70476876202052, 10.97456241420307, 11.19945735017312, 12.07324394804999, 12.29282107876363, 12.92751461089109, 13.25952775686100
