| 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 + 4·19-s + 20-s + 21-s + 8·23-s − 3·24-s + 25-s + 26-s + 27-s − 28-s + 6·29-s − 30-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 + 0.917·19-s + 0.223·20-s + 0.218·21-s + 1.66·23-s − 0.612·24-s + 1/5·25-s + 0.196·26-s + 0.192·27-s − 0.188·28-s + 1.11·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1365 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1365 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.448841450\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.448841450\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 13 | \( 1 - T \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−9.403192354398530571686930809825, −8.800181932731828826456434596625, −8.030849012237741421388049105254, −7.23869976315210479801256411617, −6.17024802827569196296941614734, −5.11340090095929037490022356539, −4.53457532457821536682054964792, −3.48385429020873155578940730375, −2.85446571166689535323654556547, −1.07211392053765584822127887766,
1.07211392053765584822127887766, 2.85446571166689535323654556547, 3.48385429020873155578940730375, 4.53457532457821536682054964792, 5.11340090095929037490022356539, 6.17024802827569196296941614734, 7.23869976315210479801256411617, 8.030849012237741421388049105254, 8.800181932731828826456434596625, 9.403192354398530571686930809825
