| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 5·7-s + 8-s − 2·9-s − 4·11-s + 12-s + 5·14-s + 16-s + 17-s − 2·18-s + 2·19-s + 5·21-s − 4·22-s − 8·23-s + 24-s − 5·27-s + 5·28-s + 5·31-s + 32-s − 4·33-s + 34-s − 2·36-s + 12·37-s + 2·38-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 1.88·7-s + 0.353·8-s − 2/3·9-s − 1.20·11-s + 0.288·12-s + 1.33·14-s + 1/4·16-s + 0.242·17-s − 0.471·18-s + 0.458·19-s + 1.09·21-s − 0.852·22-s − 1.66·23-s + 0.204·24-s − 0.962·27-s + 0.944·28-s + 0.898·31-s + 0.176·32-s − 0.696·33-s + 0.171·34-s − 1/3·36-s + 1.97·37-s + 0.324·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 143650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.218971573\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.218971573\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 \) | |
| 17 | \( 1 - T \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 7 | \( 1 - 5 T + p T^{2} \) | 1.7.af |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 - 12 T + p T^{2} \) | 1.37.am |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 + T + p T^{2} \) | 1.53.b |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 5 T + p T^{2} \) | 1.71.f |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + 17 T + p T^{2} \) | 1.79.r |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 16 T + p T^{2} \) | 1.97.q |
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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
−13.54739523607972, −13.07317541347042, −12.39882862659151, −11.88735541451945, −11.50529511415112, −11.18642662943920, −10.58999714340906, −10.15431242378187, −9.563049709809787, −8.877203883702904, −8.215033716131212, −8.084056572356247, −7.598739337344589, −7.301722472349035, −6.153594018639469, −5.828389898872477, −5.451696460194604, −4.639253358338357, −4.522785177666826, −3.818057059625753, −3.033452651528693, −2.508039459985574, −2.156136463094102, −1.437209534960206, −0.6104811852682943,
0.6104811852682943, 1.437209534960206, 2.156136463094102, 2.508039459985574, 3.033452651528693, 3.818057059625753, 4.522785177666826, 4.639253358338357, 5.451696460194604, 5.828389898872477, 6.153594018639469, 7.301722472349035, 7.598739337344589, 8.084056572356247, 8.215033716131212, 8.877203883702904, 9.563049709809787, 10.15431242378187, 10.58999714340906, 11.18642662943920, 11.50529511415112, 11.88735541451945, 12.39882862659151, 13.07317541347042, 13.54739523607972
