| L(s) = 1 | − 2-s + 4-s − 5-s − 2·7-s − 8-s + 10-s + 13-s + 2·14-s + 16-s + 4·17-s + 2·19-s − 20-s + 6·23-s + 25-s − 26-s − 2·28-s − 4·29-s − 8·31-s − 32-s − 4·34-s + 2·35-s + 2·37-s − 2·38-s + 40-s + 2·41-s + 12·43-s − 6·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.755·7-s − 0.353·8-s + 0.316·10-s + 0.277·13-s + 0.534·14-s + 1/4·16-s + 0.970·17-s + 0.458·19-s − 0.223·20-s + 1.25·23-s + 1/5·25-s − 0.196·26-s − 0.377·28-s − 0.742·29-s − 1.43·31-s − 0.176·32-s − 0.685·34-s + 0.338·35-s + 0.328·37-s − 0.324·38-s + 0.158·40-s + 0.312·41-s + 1.82·43-s − 0.884·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 141570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.789565679\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.789565679\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 4 T + p T^{2} \) | 1.97.ae |
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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
−13.17783141955715, −12.75176051089418, −12.59123106977845, −11.91132291734556, −11.28548715443447, −11.01717546700913, −10.61835464503272, −9.799802693462130, −9.542495502345720, −9.182693610861915, −8.552665528765913, −8.031176444309393, −7.520706649380977, −7.114139885783901, −6.674467626956261, −5.980067733515994, −5.489610001251396, −5.045004590572284, −4.112204951368987, −3.567816436321816, −3.229928136541348, −2.507618080012893, −1.842548336640643, −0.9079900789367524, −0.5808123241099201,
0.5808123241099201, 0.9079900789367524, 1.842548336640643, 2.507618080012893, 3.229928136541348, 3.567816436321816, 4.112204951368987, 5.045004590572284, 5.489610001251396, 5.980067733515994, 6.674467626956261, 7.114139885783901, 7.520706649380977, 8.031176444309393, 8.552665528765913, 9.182693610861915, 9.542495502345720, 9.799802693462130, 10.61835464503272, 11.01717546700913, 11.28548715443447, 11.91132291734556, 12.59123106977845, 12.75176051089418, 13.17783141955715
