| L(s) = 1 | + 5.24·2-s + 0.263·3-s + 19.4·4-s − 17.3·5-s + 1.38·6-s + 21.1·7-s + 60.2·8-s − 26.9·9-s − 90.9·10-s + 36.3·11-s + 5.13·12-s − 33.3·13-s + 110.·14-s − 4.57·15-s + 160.·16-s − 97.2·17-s − 141.·18-s − 102.·19-s − 338.·20-s + 5.57·21-s + 190.·22-s − 45.3·23-s + 15.8·24-s + 175.·25-s − 174.·26-s − 14.2·27-s + 411.·28-s + ⋯ |
| L(s) = 1 | + 1.85·2-s + 0.0507·3-s + 2.43·4-s − 1.55·5-s + 0.0940·6-s + 1.14·7-s + 2.66·8-s − 0.997·9-s − 2.87·10-s + 0.995·11-s + 0.123·12-s − 0.711·13-s + 2.11·14-s − 0.0787·15-s + 2.50·16-s − 1.38·17-s − 1.84·18-s − 1.23·19-s − 3.78·20-s + 0.0578·21-s + 1.84·22-s − 0.411·23-s + 0.135·24-s + 1.40·25-s − 1.31·26-s − 0.101·27-s + 2.78·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 43 | \( 1 \) |
| good | 2 | \( 1 - 5.24T + 8T^{2} \) |
| 3 | \( 1 - 0.263T + 27T^{2} \) |
| 5 | \( 1 + 17.3T + 125T^{2} \) |
| 7 | \( 1 - 21.1T + 343T^{2} \) |
| 11 | \( 1 - 36.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 33.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 97.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 102.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 45.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + 39.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 95.4T + 2.97e4T^{2} \) |
| 37 | \( 1 - 253.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 119.T + 6.89e4T^{2} \) |
| 47 | \( 1 + 303.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 617.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 507.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 372.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 386.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 69.4T + 3.57e5T^{2} \) |
| 73 | \( 1 + 856.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 700.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 774.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 720.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 797.T + 9.12e5T^{2} \) |
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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
−8.234963908400401866014372047593, −7.50885502927043681926232509348, −6.73030778327076128055731178921, −5.94927106483797574203452339269, −4.84287860826631801950689135380, −4.37267813808475428333232467627, −3.79840745331715517352386918851, −2.73924519730378963411392418031, −1.81381063910817629074353480381, 0,
1.81381063910817629074353480381, 2.73924519730378963411392418031, 3.79840745331715517352386918851, 4.37267813808475428333232467627, 4.84287860826631801950689135380, 5.94927106483797574203452339269, 6.73030778327076128055731178921, 7.50885502927043681926232509348, 8.234963908400401866014372047593
