| L(s) = 1 | − 2-s − 3-s − 4-s + 5-s + 6-s + 7-s + 3·8-s + 9-s − 10-s + 12-s + 13-s − 14-s − 15-s − 16-s + 2·17-s − 18-s − 20-s − 21-s − 3·24-s + 25-s − 26-s − 27-s − 28-s + 2·29-s + 30-s − 5·32-s − 2·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.377·7-s + 1.06·8-s + 1/3·9-s − 0.316·10-s + 0.288·12-s + 0.277·13-s − 0.267·14-s − 0.258·15-s − 1/4·16-s + 0.485·17-s − 0.235·18-s − 0.223·20-s − 0.218·21-s − 0.612·24-s + 1/5·25-s − 0.196·26-s − 0.192·27-s − 0.188·28-s + 0.371·29-s + 0.182·30-s − 0.883·32-s − 0.342·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 165165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.218588868\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.218588868\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 + T \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 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.ac |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
| 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
−13.24853772298861, −12.74182151576727, −12.37732535365196, −11.70881841147151, −11.17013921164297, −10.88230030906348, −10.31355523226686, −9.861811805888148, −9.487033037463499, −9.035464488488274, −8.411560359660138, −8.021169855970839, −7.533574633910711, −7.026619862030219, −6.341399494350003, −5.886318792130767, −5.334852247920819, −4.903882861795905, −4.196292928900449, −3.986156083580661, −2.965196938435446, −2.409939592432566, −1.456506093270872, −1.200697666118078, −0.4271883427745104,
0.4271883427745104, 1.200697666118078, 1.456506093270872, 2.409939592432566, 2.965196938435446, 3.986156083580661, 4.196292928900449, 4.903882861795905, 5.334852247920819, 5.886318792130767, 6.341399494350003, 7.026619862030219, 7.533574633910711, 8.021169855970839, 8.411560359660138, 9.035464488488274, 9.487033037463499, 9.861811805888148, 10.31355523226686, 10.88230030906348, 11.17013921164297, 11.70881841147151, 12.37732535365196, 12.74182151576727, 13.24853772298861
