| L(s) = 1 | + 333.·2-s − 5.66e3·3-s + 7.81e4·4-s − 5.62e4·5-s − 1.88e6·6-s + 7.14e5·7-s + 1.51e7·8-s + 1.76e7·9-s − 1.87e7·10-s − 9.95e7·11-s − 4.42e8·12-s + 9.71e6·13-s + 2.37e8·14-s + 3.18e8·15-s + 2.47e9·16-s − 1.65e9·17-s + 5.89e9·18-s − 4.03e9·19-s − 4.39e9·20-s − 4.04e9·21-s − 3.31e10·22-s + 3.40e9·23-s − 8.56e10·24-s − 2.73e10·25-s + 3.23e9·26-s − 1.88e10·27-s + 5.58e10·28-s + ⋯ |
| L(s) = 1 | + 1.84·2-s − 1.49·3-s + 2.38·4-s − 0.321·5-s − 2.74·6-s + 0.327·7-s + 2.55·8-s + 1.23·9-s − 0.592·10-s − 1.54·11-s − 3.56·12-s + 0.0429·13-s + 0.603·14-s + 0.480·15-s + 2.30·16-s − 0.975·17-s + 2.26·18-s − 1.03·19-s − 0.767·20-s − 0.489·21-s − 2.83·22-s + 0.208·23-s − 3.81·24-s − 0.896·25-s + 0.0790·26-s − 0.347·27-s + 0.782·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 - 3.40e9T \) |
| good | 2 | \( 1 - 333.T + 3.27e4T^{2} \) |
| 3 | \( 1 + 5.66e3T + 1.43e7T^{2} \) |
| 5 | \( 1 + 5.62e4T + 3.05e10T^{2} \) |
| 7 | \( 1 - 7.14e5T + 4.74e12T^{2} \) |
| 11 | \( 1 + 9.95e7T + 4.17e15T^{2} \) |
| 13 | \( 1 - 9.71e6T + 5.11e16T^{2} \) |
| 17 | \( 1 + 1.65e9T + 2.86e18T^{2} \) |
| 19 | \( 1 + 4.03e9T + 1.51e19T^{2} \) |
| 29 | \( 1 + 1.48e11T + 8.62e21T^{2} \) |
| 31 | \( 1 + 1.73e11T + 2.34e22T^{2} \) |
| 37 | \( 1 - 9.32e11T + 3.33e23T^{2} \) |
| 41 | \( 1 + 1.07e12T + 1.55e24T^{2} \) |
| 43 | \( 1 - 1.58e12T + 3.17e24T^{2} \) |
| 47 | \( 1 - 5.31e12T + 1.20e25T^{2} \) |
| 53 | \( 1 - 9.86e12T + 7.31e25T^{2} \) |
| 59 | \( 1 + 1.47e13T + 3.65e26T^{2} \) |
| 61 | \( 1 + 2.01e13T + 6.02e26T^{2} \) |
| 67 | \( 1 - 4.07e13T + 2.46e27T^{2} \) |
| 71 | \( 1 - 1.17e14T + 5.87e27T^{2} \) |
| 73 | \( 1 + 4.58e13T + 8.90e27T^{2} \) |
| 79 | \( 1 + 1.72e14T + 2.91e28T^{2} \) |
| 83 | \( 1 - 1.04e14T + 6.11e28T^{2} \) |
| 89 | \( 1 - 3.72e14T + 1.74e29T^{2} \) |
| 97 | \( 1 + 8.00e14T + 6.33e29T^{2} \) |
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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
−13.30344737787643169786469016106, −12.56209244250857872162842767004, −11.28986146299849248074222394379, −10.78608285955136122961120848924, −7.44328414315327162105739411989, −6.05976004505753870010347303421, −5.19336774724483490167742035609, −4.13626958249507163214625092450, −2.21934105502287138311578334058, 0,
2.21934105502287138311578334058, 4.13626958249507163214625092450, 5.19336774724483490167742035609, 6.05976004505753870010347303421, 7.44328414315327162105739411989, 10.78608285955136122961120848924, 11.28986146299849248074222394379, 12.56209244250857872162842767004, 13.30344737787643169786469016106
