| L(s) = 1 | + 2·5-s + 7-s − 2·11-s + 2·13-s + 6·17-s + 4·19-s − 6·23-s − 25-s + 4·31-s + 2·35-s + 10·37-s + 2·41-s + 4·43-s − 4·47-s + 49-s − 12·53-s − 4·55-s − 12·59-s + 6·61-s + 4·65-s + 4·67-s + 14·71-s − 2·73-s − 2·77-s + 8·79-s + 16·83-s + 12·85-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.377·7-s − 0.603·11-s + 0.554·13-s + 1.45·17-s + 0.917·19-s − 1.25·23-s − 1/5·25-s + 0.718·31-s + 0.338·35-s + 1.64·37-s + 0.312·41-s + 0.609·43-s − 0.583·47-s + 1/7·49-s − 1.64·53-s − 0.539·55-s − 1.56·59-s + 0.768·61-s + 0.496·65-s + 0.488·67-s + 1.66·71-s − 0.234·73-s − 0.227·77-s + 0.900·79-s + 1.75·83-s + 1.30·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.021268337\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.021268337\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 3 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 14 T + p T^{2} \) | 1.71.ao |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−9.818151445135231115814034817807, −9.437752301009204937027256546920, −8.016710564482448925343947507213, −7.79064855464535275520223516342, −6.32147524527224537125089512133, −5.73569194086517270564919024713, −4.88053771103235633099685381109, −3.60698435135707065201863764202, −2.45683901738953215093540180120, −1.23073197938907604520302484273,
1.23073197938907604520302484273, 2.45683901738953215093540180120, 3.60698435135707065201863764202, 4.88053771103235633099685381109, 5.73569194086517270564919024713, 6.32147524527224537125089512133, 7.79064855464535275520223516342, 8.016710564482448925343947507213, 9.437752301009204937027256546920, 9.818151445135231115814034817807
