| L(s) = 1 | − 2·2-s − 3·3-s + 4·4-s + 10.1·5-s + 6·6-s − 32.1·7-s − 8·8-s + 9·9-s − 20.3·10-s + 60.6·11-s − 12·12-s + 35.4·13-s + 64.3·14-s − 30.5·15-s + 16·16-s + 8.87·17-s − 18·18-s + 40.6·20-s + 96.4·21-s − 121.·22-s − 50.2·23-s + 24·24-s − 21.6·25-s − 70.8·26-s − 27·27-s − 128.·28-s − 268.·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.909·5-s + 0.408·6-s − 1.73·7-s − 0.353·8-s + 0.333·9-s − 0.643·10-s + 1.66·11-s − 0.288·12-s + 0.755·13-s + 1.22·14-s − 0.525·15-s + 0.250·16-s + 0.126·17-s − 0.235·18-s + 0.454·20-s + 1.00·21-s − 1.17·22-s − 0.455·23-s + 0.204·24-s − 0.172·25-s − 0.534·26-s − 0.192·27-s − 0.868·28-s − 1.71·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + 2T \) |
| 3 | \( 1 + 3T \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 - 10.1T + 125T^{2} \) |
| 7 | \( 1 + 32.1T + 343T^{2} \) |
| 11 | \( 1 - 60.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 35.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 8.87T + 4.91e3T^{2} \) |
| 23 | \( 1 + 50.2T + 1.21e4T^{2} \) |
| 29 | \( 1 + 268.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 68.9T + 2.97e4T^{2} \) |
| 37 | \( 1 + 197.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 298.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 315.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 162.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 583.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 416.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 501.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 294.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 885.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 795.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 212.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 557.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 336.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.86e3T + 9.12e5T^{2} \) |
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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
−8.662852845549866471457353297576, −7.31109959195940361638333279352, −6.66038073621651008518247280501, −6.08414073659376384774872934657, −5.65847121508858964110165237032, −4.01198000015665445483505279784, −3.37644728046384907619478858919, −2.04861758107921316541475233872, −1.12237282128968521283663533351, 0,
1.12237282128968521283663533351, 2.04861758107921316541475233872, 3.37644728046384907619478858919, 4.01198000015665445483505279784, 5.65847121508858964110165237032, 6.08414073659376384774872934657, 6.66038073621651008518247280501, 7.31109959195940361638333279352, 8.662852845549866471457353297576
