| L(s) = 1 | − 5·5-s + 14.7·7-s − 7.25·11-s + 61.7·13-s + 108.·17-s + 56.2·19-s + 46.9·23-s + 25·25-s − 214.·29-s + 261.·31-s − 73.7·35-s − 286·37-s + 255.·41-s + 361.·43-s + 5.53·47-s − 125.·49-s − 595.·53-s + 36.2·55-s − 315.·59-s + 276.·61-s − 308.·65-s − 117.·67-s − 192.·71-s − 756.·73-s − 106.·77-s + 1.15e3·79-s − 141.·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.796·7-s − 0.198·11-s + 1.31·13-s + 1.54·17-s + 0.679·19-s + 0.425·23-s + 0.200·25-s − 1.37·29-s + 1.51·31-s − 0.356·35-s − 1.27·37-s + 0.973·41-s + 1.28·43-s + 0.0171·47-s − 0.365·49-s − 1.54·53-s + 0.0888·55-s − 0.695·59-s + 0.580·61-s − 0.589·65-s − 0.214·67-s − 0.322·71-s − 1.21·73-s − 0.158·77-s + 1.63·79-s − 0.187·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.736732064\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.736732064\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| good | 7 | \( 1 - 14.7T + 343T^{2} \) |
| 11 | \( 1 + 7.25T + 1.33e3T^{2} \) |
| 13 | \( 1 - 61.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 108.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 56.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 46.9T + 1.21e4T^{2} \) |
| 29 | \( 1 + 214.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 261.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 286T + 5.06e4T^{2} \) |
| 41 | \( 1 - 255.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 361.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 5.53T + 1.03e5T^{2} \) |
| 53 | \( 1 + 595.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 315.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 276.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 117.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 192.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 756.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.15e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 141.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 719.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.04e3T + 9.12e5T^{2} \) |
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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
−8.628442886736332769039368087324, −7.83580942002962935874840310454, −7.48002417380488389910445260408, −6.27366145499345130168390591261, −5.53745967154544629117673218131, −4.73258208864253161497545598446, −3.73650548552533821513075148450, −3.03143125922218684439054559202, −1.60705514738449979110797926785, −0.815837609785272161945013762832,
0.815837609785272161945013762832, 1.60705514738449979110797926785, 3.03143125922218684439054559202, 3.73650548552533821513075148450, 4.73258208864253161497545598446, 5.53745967154544629117673218131, 6.27366145499345130168390591261, 7.48002417380488389910445260408, 7.83580942002962935874840310454, 8.628442886736332769039368087324
