| L(s) = 1 | − 1.67·2-s − 8.29·3-s − 5.20·4-s + 3.53·5-s + 13.8·6-s + 22.0·8-s + 41.7·9-s − 5.91·10-s − 44.0·11-s + 43.1·12-s − 48.8·13-s − 29.2·15-s + 4.65·16-s − 65.5·17-s − 69.8·18-s + 16.8·19-s − 18.3·20-s + 73.7·22-s + 10.1·23-s − 183.·24-s − 112.·25-s + 81.7·26-s − 122.·27-s − 107.·29-s + 49.0·30-s + 98.8·31-s − 184.·32-s + ⋯ |
| L(s) = 1 | − 0.591·2-s − 1.59·3-s − 0.650·4-s + 0.316·5-s + 0.943·6-s + 0.976·8-s + 1.54·9-s − 0.186·10-s − 1.20·11-s + 1.03·12-s − 1.04·13-s − 0.504·15-s + 0.0727·16-s − 0.935·17-s − 0.914·18-s + 0.203·19-s − 0.205·20-s + 0.714·22-s + 0.0916·23-s − 1.55·24-s − 0.900·25-s + 0.616·26-s − 0.870·27-s − 0.690·29-s + 0.298·30-s + 0.572·31-s − 1.01·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2303 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2303 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.01286190291\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01286190291\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 47 | \( 1 - 47T \) |
| good | 2 | \( 1 + 1.67T + 8T^{2} \) |
| 3 | \( 1 + 8.29T + 27T^{2} \) |
| 5 | \( 1 - 3.53T + 125T^{2} \) |
| 11 | \( 1 + 44.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 48.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 65.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 16.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 10.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + 107.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 98.8T + 2.97e4T^{2} \) |
| 37 | \( 1 + 68.8T + 5.06e4T^{2} \) |
| 41 | \( 1 - 224.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 126.T + 7.95e4T^{2} \) |
| 53 | \( 1 - 164.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 159.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 686.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 894.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.04e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 303.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 444.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.24e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.20e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 167.T + 9.12e5T^{2} \) |
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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
−8.794478665894193005263425104888, −7.68203330882893365587471747537, −7.28356963975090287206066965342, −6.14620245341810505623610728755, −5.48822658263774848593210527794, −4.83370824420151731501644740975, −4.21893585059824767730332947351, −2.57418882733755395131830373115, −1.37817291089965896374130045351, −0.06450415108690328366192210242,
0.06450415108690328366192210242, 1.37817291089965896374130045351, 2.57418882733755395131830373115, 4.21893585059824767730332947351, 4.83370824420151731501644740975, 5.48822658263774848593210527794, 6.14620245341810505623610728755, 7.28356963975090287206066965342, 7.68203330882893365587471747537, 8.794478665894193005263425104888
