| L(s) = 1 | + 5-s − 2·7-s − 3·9-s + 8·19-s + 8·23-s + 25-s + 10·29-s − 8·31-s − 2·35-s + 10·37-s + 2·41-s + 6·43-s − 3·45-s + 8·47-s − 3·49-s − 14·53-s − 4·59-s + 10·61-s + 6·63-s + 4·67-s + 8·73-s − 4·79-s + 9·81-s − 10·83-s + 6·89-s + 8·95-s − 10·97-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.755·7-s − 9-s + 1.83·19-s + 1.66·23-s + 1/5·25-s + 1.85·29-s − 1.43·31-s − 0.338·35-s + 1.64·37-s + 0.312·41-s + 0.914·43-s − 0.447·45-s + 1.16·47-s − 3/7·49-s − 1.92·53-s − 0.520·59-s + 1.28·61-s + 0.755·63-s + 0.488·67-s + 0.936·73-s − 0.450·79-s + 81-s − 1.09·83-s + 0.635·89-s + 0.820·95-s − 1.01·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.465812132\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.465812132\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 14 T + p T^{2} \) | 1.53.o |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 10 T + p T^{2} \) | 1.83.k |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
| 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
−14.57700544204270, −14.32003416357321, −13.88251883547191, −13.23060856633305, −12.77380878419759, −12.32657332264470, −11.59757986137552, −11.13970316851272, −10.74164895364978, −9.896316223709346, −9.486512274221251, −9.109920946946469, −8.529864291079791, −7.742775340863794, −7.317903197901654, −6.537063975979399, −6.168209967823882, −5.389292326079878, −5.129237082698562, −4.248359501682824, −3.312047148769456, −2.955554321537895, −2.437252888301206, −1.231038508683612, −0.6404986689455263,
0.6404986689455263, 1.231038508683612, 2.437252888301206, 2.955554321537895, 3.312047148769456, 4.248359501682824, 5.129237082698562, 5.389292326079878, 6.168209967823882, 6.537063975979399, 7.317903197901654, 7.742775340863794, 8.529864291079791, 9.109920946946469, 9.486512274221251, 9.896316223709346, 10.74164895364978, 11.13970316851272, 11.59757986137552, 12.32657332264470, 12.77380878419759, 13.23060856633305, 13.88251883547191, 14.32003416357321, 14.57700544204270
