| L(s) = 1 | + 3-s + 4·7-s + 9-s − 13-s − 2·17-s + 4·21-s + 27-s − 2·29-s + 4·31-s − 6·37-s − 39-s − 6·41-s + 4·43-s − 4·47-s + 9·49-s − 2·51-s + 10·53-s − 2·61-s + 4·63-s + 8·67-s − 4·71-s + 6·73-s + 8·79-s + 81-s + 8·83-s − 2·87-s − 6·89-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.51·7-s + 1/3·9-s − 0.277·13-s − 0.485·17-s + 0.872·21-s + 0.192·27-s − 0.371·29-s + 0.718·31-s − 0.986·37-s − 0.160·39-s − 0.937·41-s + 0.609·43-s − 0.583·47-s + 9/7·49-s − 0.280·51-s + 1.37·53-s − 0.256·61-s + 0.503·63-s + 0.977·67-s − 0.474·71-s + 0.702·73-s + 0.900·79-s + 1/9·81-s + 0.878·83-s − 0.214·87-s − 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.398141163\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.398141163\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.77042927054710, −15.36739050820556, −14.86571036299748, −14.33380162878176, −13.88905316409298, −13.37547336435022, −12.68285857923022, −11.98391205117783, −11.55331487664260, −10.93802807553898, −10.35294868530313, −9.776861006474280, −8.910523300912727, −8.588238430439868, −7.965392872211431, −7.432623573471863, −6.824634117623842, −5.995531874658153, −5.096443030580759, −4.768176461356187, −3.987712622935149, −3.249241251665876, −2.226790042866871, −1.832048734941535, −0.7839049613117447,
0.7839049613117447, 1.832048734941535, 2.226790042866871, 3.249241251665876, 3.987712622935149, 4.768176461356187, 5.096443030580759, 5.995531874658153, 6.824634117623842, 7.432623573471863, 7.965392872211431, 8.588238430439868, 8.910523300912727, 9.776861006474280, 10.35294868530313, 10.93802807553898, 11.55331487664260, 11.98391205117783, 12.68285857923022, 13.37547336435022, 13.88905316409298, 14.33380162878176, 14.86571036299748, 15.36739050820556, 15.77042927054710
