| L(s) = 1 | + 2·7-s − 3·9-s − 4·11-s + 6·13-s + 4·17-s + 4·19-s − 6·23-s − 29-s + 8·37-s − 2·41-s − 4·43-s + 4·47-s − 3·49-s + 2·53-s + 8·59-s + 10·61-s − 6·63-s + 10·67-s − 8·71-s − 8·77-s + 8·79-s + 9·81-s + 6·83-s + 6·89-s + 12·91-s + 12·97-s + 12·99-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 9-s − 1.20·11-s + 1.66·13-s + 0.970·17-s + 0.917·19-s − 1.25·23-s − 0.185·29-s + 1.31·37-s − 0.312·41-s − 0.609·43-s + 0.583·47-s − 3/7·49-s + 0.274·53-s + 1.04·59-s + 1.28·61-s − 0.755·63-s + 1.22·67-s − 0.949·71-s − 0.911·77-s + 0.900·79-s + 81-s + 0.658·83-s + 0.635·89-s + 1.25·91-s + 1.21·97-s + 1.20·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.877332671\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.877332671\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 29 | \( 1 + T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + p T^{2} \) | 1.73.a |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
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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
−8.437982612948757466144236967999, −8.166344273072527605155220037306, −7.52041515223519037401787250034, −6.28804970205735497830422095583, −5.62139491694076595892476908285, −5.12393318973471119896749759750, −3.90459166688862309008139341358, −3.13872298844776834415772073961, −2.10194821002577286786817841759, −0.856899419421507899299452669904,
0.856899419421507899299452669904, 2.10194821002577286786817841759, 3.13872298844776834415772073961, 3.90459166688862309008139341358, 5.12393318973471119896749759750, 5.62139491694076595892476908285, 6.28804970205735497830422095583, 7.52041515223519037401787250034, 8.166344273072527605155220037306, 8.437982612948757466144236967999
