| L(s) = 1 | − 2·2-s + 3·3-s + 4·4-s − 19.5·5-s − 6·6-s + 7·7-s − 8·8-s + 9·9-s + 39.1·10-s + 3.18·11-s + 12·12-s + 13·13-s − 14·14-s − 58.7·15-s + 16·16-s + 51.5·17-s − 18·18-s + 36.7·19-s − 78.3·20-s + 21·21-s − 6.37·22-s − 7.21·23-s − 24·24-s + 258.·25-s − 26·26-s + 27·27-s + 28·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.75·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s + 1.23·10-s + 0.0873·11-s + 0.288·12-s + 0.277·13-s − 0.267·14-s − 1.01·15-s + 0.250·16-s + 0.735·17-s − 0.235·18-s + 0.444·19-s − 0.876·20-s + 0.218·21-s − 0.0617·22-s − 0.0654·23-s − 0.204·24-s + 2.07·25-s − 0.196·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 3T \) |
| 7 | \( 1 - 7T \) |
| 13 | \( 1 - 13T \) |
| good | 5 | \( 1 + 19.5T + 125T^{2} \) |
| 11 | \( 1 - 3.18T + 1.33e3T^{2} \) |
| 17 | \( 1 - 51.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 36.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 7.21T + 1.21e4T^{2} \) |
| 29 | \( 1 + 133.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 0.780T + 2.97e4T^{2} \) |
| 37 | \( 1 + 187.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 246.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 222.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 438.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 153.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 336T + 2.05e5T^{2} \) |
| 61 | \( 1 - 230T + 2.26e5T^{2} \) |
| 67 | \( 1 - 898.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 584.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 902.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 157.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 457.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 171.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 926.T + 9.12e5T^{2} \) |
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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
−9.878874681283250775365925688021, −8.819607067310623916559737912849, −8.129371455298772401010818205966, −7.59693139191234690543688202557, −6.74747017171572877679856275635, −5.12948045312877290552638972440, −3.88022558769286366939280243822, −3.13353101154733424064332874237, −1.43696578685111072430136916416, 0,
1.43696578685111072430136916416, 3.13353101154733424064332874237, 3.88022558769286366939280243822, 5.12948045312877290552638972440, 6.74747017171572877679856275635, 7.59693139191234690543688202557, 8.129371455298772401010818205966, 8.819607067310623916559737912849, 9.878874681283250775365925688021
