| L(s) = 1 | − 2·3-s + 2·5-s + 4·7-s + 9-s + 2·11-s + 2·13-s − 4·15-s − 2·17-s − 2·19-s − 8·21-s − 4·23-s − 25-s + 4·27-s − 6·29-s − 4·33-s + 8·35-s + 10·37-s − 4·39-s − 6·41-s − 6·43-s + 2·45-s + 8·47-s + 9·49-s + 4·51-s − 6·53-s + 4·55-s + 4·57-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s + 1.51·7-s + 1/3·9-s + 0.603·11-s + 0.554·13-s − 1.03·15-s − 0.485·17-s − 0.458·19-s − 1.74·21-s − 0.834·23-s − 1/5·25-s + 0.769·27-s − 1.11·29-s − 0.696·33-s + 1.35·35-s + 1.64·37-s − 0.640·39-s − 0.937·41-s − 0.914·43-s + 0.298·45-s + 1.16·47-s + 9/7·49-s + 0.560·51-s − 0.824·53-s + 0.539·55-s + 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9713362299\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9713362299\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 14 T + p T^{2} \) | 1.59.o |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 10 T + p T^{2} \) | 1.67.k |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 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
−13.38742730478456157071741394596, −12.03375355269583309070551087085, −11.28853583105345723449818824184, −10.57265847830629711938727340297, −9.215939702261382322141825805489, −7.984207807149970007419508597610, −6.39871498618546548325782647590, −5.59100833138318705450052051530, −4.43324404972048918127214240916, −1.73703685485230483844290438322,
1.73703685485230483844290438322, 4.43324404972048918127214240916, 5.59100833138318705450052051530, 6.39871498618546548325782647590, 7.984207807149970007419508597610, 9.215939702261382322141825805489, 10.57265847830629711938727340297, 11.28853583105345723449818824184, 12.03375355269583309070551087085, 13.38742730478456157071741394596
