| L(s) = 1 | − 0.466·3-s − 6.69·5-s + 7·7-s − 26.7·9-s + 11·11-s + 57.3·13-s + 3.12·15-s − 7.31·17-s + 111.·19-s − 3.26·21-s − 113.·23-s − 80.1·25-s + 25.0·27-s + 51.2·29-s − 256.·31-s − 5.13·33-s − 46.8·35-s − 11.5·37-s − 26.7·39-s + 57.3·41-s − 559.·43-s + 179.·45-s + 110.·47-s + 49·49-s + 3.41·51-s + 631.·53-s − 73.6·55-s + ⋯ |
| L(s) = 1 | − 0.0898·3-s − 0.599·5-s + 0.377·7-s − 0.991·9-s + 0.301·11-s + 1.22·13-s + 0.0538·15-s − 0.104·17-s + 1.34·19-s − 0.0339·21-s − 1.02·23-s − 0.640·25-s + 0.178·27-s + 0.328·29-s − 1.48·31-s − 0.0270·33-s − 0.226·35-s − 0.0513·37-s − 0.109·39-s + 0.218·41-s − 1.98·43-s + 0.594·45-s + 0.343·47-s + 0.142·49-s + 0.00936·51-s + 1.63·53-s − 0.180·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.624006175\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.624006175\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| 11 | \( 1 - 11T \) |
| good | 3 | \( 1 + 0.466T + 27T^{2} \) |
| 5 | \( 1 + 6.69T + 125T^{2} \) |
| 13 | \( 1 - 57.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 7.31T + 4.91e3T^{2} \) |
| 19 | \( 1 - 111.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 113.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 51.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 256.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 11.5T + 5.06e4T^{2} \) |
| 41 | \( 1 - 57.3T + 6.89e4T^{2} \) |
| 43 | \( 1 + 559.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 110.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 631.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 590.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 9.24T + 2.26e5T^{2} \) |
| 67 | \( 1 - 600.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 842.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 734.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 737.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 265.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 909.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.65e3T + 9.12e5T^{2} \) |
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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
−9.200928114856947650156700936748, −8.442330738740371309543084950787, −7.87320546336657434135811672532, −6.89369745486295468780273209498, −5.87953160317019407507183950956, −5.24655477306927844590063045838, −3.94382895927978185697756710197, −3.35437390901574907664249860263, −1.94516317992995127595829630777, −0.65289168032938232957733969051,
0.65289168032938232957733969051, 1.94516317992995127595829630777, 3.35437390901574907664249860263, 3.94382895927978185697756710197, 5.24655477306927844590063045838, 5.87953160317019407507183950956, 6.89369745486295468780273209498, 7.87320546336657434135811672532, 8.442330738740371309543084950787, 9.200928114856947650156700936748
