| L(s) = 1 | − 4.83·2-s − 7.08·3-s + 15.3·4-s + 9.07·5-s + 34.2·6-s + 2.15·7-s − 35.4·8-s + 23.2·9-s − 43.8·10-s − 11·11-s − 108.·12-s − 10.4·14-s − 64.3·15-s + 48.4·16-s + 49.8·17-s − 112.·18-s + 95.3·19-s + 139.·20-s − 15.2·21-s + 53.1·22-s − 96.0·23-s + 251.·24-s − 42.6·25-s + 26.6·27-s + 33.0·28-s + 115.·29-s + 310.·30-s + ⋯ |
| L(s) = 1 | − 1.70·2-s − 1.36·3-s + 1.91·4-s + 0.811·5-s + 2.32·6-s + 0.116·7-s − 1.56·8-s + 0.860·9-s − 1.38·10-s − 0.301·11-s − 2.61·12-s − 0.198·14-s − 1.10·15-s + 0.756·16-s + 0.711·17-s − 1.47·18-s + 1.15·19-s + 1.55·20-s − 0.158·21-s + 0.514·22-s − 0.871·23-s + 2.13·24-s − 0.341·25-s + 0.189·27-s + 0.223·28-s + 0.736·29-s + 1.89·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + 11T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 4.83T + 8T^{2} \) |
| 3 | \( 1 + 7.08T + 27T^{2} \) |
| 5 | \( 1 - 9.07T + 125T^{2} \) |
| 7 | \( 1 - 2.15T + 343T^{2} \) |
| 17 | \( 1 - 49.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 95.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 96.0T + 1.21e4T^{2} \) |
| 29 | \( 1 - 115.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 78.6T + 2.97e4T^{2} \) |
| 37 | \( 1 + 13.4T + 5.06e4T^{2} \) |
| 41 | \( 1 + 122.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 95.5T + 7.95e4T^{2} \) |
| 47 | \( 1 + 361.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 617.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 723.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 549.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 554.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 99.2T + 3.57e5T^{2} \) |
| 73 | \( 1 + 609.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 586.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 5.23T + 5.71e5T^{2} \) |
| 89 | \( 1 + 789.T + 7.04e5T^{2} \) |
| 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
−8.515263003155460836581071162364, −7.78397921241041712740274353439, −6.99510760539184346282375633562, −6.16014872786719887786352392164, −5.64480975434470692685012444863, −4.69803408327296558717899571023, −3.00034100339696436288970918688, −1.78289929543610708464300869293, −0.988319188236137167928241106737, 0,
0.988319188236137167928241106737, 1.78289929543610708464300869293, 3.00034100339696436288970918688, 4.69803408327296558717899571023, 5.64480975434470692685012444863, 6.16014872786719887786352392164, 6.99510760539184346282375633562, 7.78397921241041712740274353439, 8.515263003155460836581071162364
