| L(s) = 1 | + 2-s + 3-s − 4-s + 6-s − 7-s − 3·8-s + 9-s − 6·11-s − 12-s + 13-s − 14-s − 16-s + 18-s + 2·19-s − 21-s − 6·22-s − 3·24-s + 26-s + 27-s + 28-s + 6·29-s − 2·31-s + 5·32-s − 6·33-s − 36-s − 8·37-s + 2·38-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.408·6-s − 0.377·7-s − 1.06·8-s + 1/3·9-s − 1.80·11-s − 0.288·12-s + 0.277·13-s − 0.267·14-s − 1/4·16-s + 0.235·18-s + 0.458·19-s − 0.218·21-s − 1.27·22-s − 0.612·24-s + 0.196·26-s + 0.192·27-s + 0.188·28-s + 1.11·29-s − 0.359·31-s + 0.883·32-s − 1.04·33-s − 1/6·36-s − 1.31·37-s + 0.324·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.032533507\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.032533507\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 - T \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 14 T + p T^{2} \) | 1.71.o |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−7.936309964584622452691307181640, −7.39192877625088412605451876968, −6.43352235144913402633276474579, −5.66051722429490033363236284528, −5.06860747191705000287079464944, −4.42675251957163083908355608365, −3.45934498595500345357030587654, −2.99911947823485710854032736465, −2.16233599751962324679151043584, −0.61329912827667362407871541867,
0.61329912827667362407871541867, 2.16233599751962324679151043584, 2.99911947823485710854032736465, 3.45934498595500345357030587654, 4.42675251957163083908355608365, 5.06860747191705000287079464944, 5.66051722429490033363236284528, 6.43352235144913402633276474579, 7.39192877625088412605451876968, 7.936309964584622452691307181640
