| L(s) = 1 | − 2·2-s + 3.03·3-s + 4·4-s − 14.4·5-s − 6.06·6-s + 31.4·7-s − 8·8-s − 17.8·9-s + 28.9·10-s − 42.7·11-s + 12.1·12-s − 32.9·13-s − 62.9·14-s − 43.8·15-s + 16·16-s − 27.3·17-s + 35.6·18-s − 148.·19-s − 57.8·20-s + 95.4·21-s + 85.5·22-s + 89.7·23-s − 24.2·24-s + 84.2·25-s + 65.9·26-s − 135.·27-s + 125.·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.583·3-s + 0.5·4-s − 1.29·5-s − 0.412·6-s + 1.69·7-s − 0.353·8-s − 0.659·9-s + 0.914·10-s − 1.17·11-s + 0.291·12-s − 0.703·13-s − 1.20·14-s − 0.755·15-s + 0.250·16-s − 0.389·17-s + 0.466·18-s − 1.79·19-s − 0.646·20-s + 0.991·21-s + 0.829·22-s + 0.813·23-s − 0.206·24-s + 0.673·25-s + 0.497·26-s − 0.968·27-s + 0.849·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1922 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1922 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.7789609317\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7789609317\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 31 | \( 1 \) |
| good | 3 | \( 1 - 3.03T + 27T^{2} \) |
| 5 | \( 1 + 14.4T + 125T^{2} \) |
| 7 | \( 1 - 31.4T + 343T^{2} \) |
| 11 | \( 1 + 42.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 32.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 27.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 148.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 89.7T + 1.21e4T^{2} \) |
| 29 | \( 1 + 179.T + 2.43e4T^{2} \) |
| 37 | \( 1 - 100.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 336.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 532.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 33.1T + 1.03e5T^{2} \) |
| 53 | \( 1 + 416.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 116.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 139.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 380.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.01e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 385.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 469.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 418.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 464.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 712.T + 9.12e5T^{2} \) |
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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
−8.581220667972739493773121687870, −7.955328056063332100555333112899, −7.83889206904895579038831583191, −6.88395458524861797019755957434, −5.50452322079603361183843564500, −4.72252562058911380526777014597, −3.85483601688751392534637440947, −2.61293588478111990091797386091, −1.98399911868277081471765446431, −0.41900476607632897040095932801,
0.41900476607632897040095932801, 1.98399911868277081471765446431, 2.61293588478111990091797386091, 3.85483601688751392534637440947, 4.72252562058911380526777014597, 5.50452322079603361183843564500, 6.88395458524861797019755957434, 7.83889206904895579038831583191, 7.955328056063332100555333112899, 8.581220667972739493773121687870
