| L(s) = 1 | − 3·3-s + 14.1·5-s + 7·7-s + 9·9-s + 11.8·11-s − 22.6·13-s − 42.4·15-s + 79.1·17-s + 55.6·19-s − 21·21-s − 84.8·23-s + 75.6·25-s − 27·27-s + 117.·29-s + 126.·31-s − 35.5·33-s + 99.1·35-s + 201.·37-s + 67.9·39-s − 10.4·41-s + 2.64·43-s + 127.·45-s − 494.·47-s + 49·49-s − 237.·51-s + 763.·53-s + 167.·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.26·5-s + 0.377·7-s + 0.333·9-s + 0.324·11-s − 0.483·13-s − 0.731·15-s + 1.12·17-s + 0.672·19-s − 0.218·21-s − 0.769·23-s + 0.605·25-s − 0.192·27-s + 0.749·29-s + 0.735·31-s − 0.187·33-s + 0.478·35-s + 0.894·37-s + 0.279·39-s − 0.0399·41-s + 0.00939·43-s + 0.422·45-s − 1.53·47-s + 0.142·49-s − 0.652·51-s + 1.97·53-s + 0.410·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.625825647\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.625825647\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 7 | \( 1 - 7T \) |
| good | 5 | \( 1 - 14.1T + 125T^{2} \) |
| 11 | \( 1 - 11.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 22.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 79.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 55.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + 84.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 117.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 126.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 201.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 10.4T + 6.89e4T^{2} \) |
| 43 | \( 1 - 2.64T + 7.95e4T^{2} \) |
| 47 | \( 1 + 494.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 763.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 55.6T + 2.05e5T^{2} \) |
| 61 | \( 1 - 457.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 291.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 956.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 517.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 830.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 346.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 584.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 85.6T + 9.12e5T^{2} \) |
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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
−9.580423142865685817753041750476, −8.437597961300833821338464027996, −7.55823323190234632958276103890, −6.60993620844506709502327966521, −5.82569944985778945614616746741, −5.25926446855427186830728274270, −4.29058038974094302428524818125, −2.93735240922208278933864530586, −1.82359035266362216467776605318, −0.881558221599682429437148180137,
0.881558221599682429437148180137, 1.82359035266362216467776605318, 2.93735240922208278933864530586, 4.29058038974094302428524818125, 5.25926446855427186830728274270, 5.82569944985778945614616746741, 6.60993620844506709502327966521, 7.55823323190234632958276103890, 8.437597961300833821338464027996, 9.580423142865685817753041750476
