L(s) = 1 | − 2-s + 4-s + 2·5-s − 2·7-s − 8-s − 3·9-s − 2·10-s + 11-s − 6·13-s + 2·14-s + 16-s + 3·18-s − 19-s + 2·20-s − 22-s + 5·23-s − 25-s + 6·26-s − 2·28-s + 6·29-s − 5·31-s − 32-s − 4·35-s − 3·36-s − 7·37-s + 38-s − 2·40-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.755·7-s − 0.353·8-s − 9-s − 0.632·10-s + 0.301·11-s − 1.66·13-s + 0.534·14-s + 1/4·16-s + 0.707·18-s − 0.229·19-s + 0.447·20-s − 0.213·22-s + 1.04·23-s − 1/5·25-s + 1.17·26-s − 0.377·28-s + 1.11·29-s − 0.898·31-s − 0.176·32-s − 0.676·35-s − 1/2·36-s − 1.15·37-s + 0.162·38-s − 0.316·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.004650738\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.004650738\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 2011 | \( 1 - T \) |
good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 13 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−8.758514660958512456172076583084, −7.67996912850603545163901743779, −6.98371753161093613766924459166, −6.36659611934092783344053929206, −5.56797270528988782450887407896, −4.94804951453696901773037818879, −3.57550812975576767990438489452, −2.65505002782742736404434308279, −2.10139511056849053935342562507, −0.60495245638741506584908823431,
0.60495245638741506584908823431, 2.10139511056849053935342562507, 2.65505002782742736404434308279, 3.57550812975576767990438489452, 4.94804951453696901773037818879, 5.56797270528988782450887407896, 6.36659611934092783344053929206, 6.98371753161093613766924459166, 7.67996912850603545163901743779, 8.758514660958512456172076583084