L(s) = 1 | − 3.42·2-s − 3·3-s + 3.72·4-s − 0.330·5-s + 10.2·6-s − 34.8·7-s + 14.6·8-s + 9·9-s + 1.13·10-s − 11·11-s − 11.1·12-s + 25.0·13-s + 119.·14-s + 0.990·15-s − 79.9·16-s − 71.1·17-s − 30.8·18-s + 74.1·19-s − 1.23·20-s + 104.·21-s + 37.6·22-s + 13.0·23-s − 43.8·24-s − 124.·25-s − 85.7·26-s − 27·27-s − 130.·28-s + ⋯ |
L(s) = 1 | − 1.21·2-s − 0.577·3-s + 0.466·4-s − 0.0295·5-s + 0.699·6-s − 1.88·7-s + 0.646·8-s + 0.333·9-s + 0.0357·10-s − 0.301·11-s − 0.269·12-s + 0.534·13-s + 2.27·14-s + 0.0170·15-s − 1.24·16-s − 1.01·17-s − 0.403·18-s + 0.895·19-s − 0.0137·20-s + 1.08·21-s + 0.365·22-s + 0.117·23-s − 0.373·24-s − 0.999·25-s − 0.647·26-s − 0.192·27-s − 0.877·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + 3T \) |
| 11 | \( 1 + 11T \) |
| 61 | \( 1 - 61T \) |
good | 2 | \( 1 + 3.42T + 8T^{2} \) |
| 5 | \( 1 + 0.330T + 125T^{2} \) |
| 7 | \( 1 + 34.8T + 343T^{2} \) |
| 13 | \( 1 - 25.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 71.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 74.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 13.0T + 1.21e4T^{2} \) |
| 29 | \( 1 + 107.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 21.5T + 2.97e4T^{2} \) |
| 37 | \( 1 + 32.4T + 5.06e4T^{2} \) |
| 41 | \( 1 - 11.4T + 6.89e4T^{2} \) |
| 43 | \( 1 - 67.7T + 7.95e4T^{2} \) |
| 47 | \( 1 + 283.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 234.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 315.T + 2.05e5T^{2} \) |
| 67 | \( 1 + 902.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 742.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 531.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.35e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.44e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 495.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 605.T + 9.12e5T^{2} \) |
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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
−8.601117620805056390712187984724, −7.62129537969286697841465014344, −6.91768835792201796290064241342, −6.27760163357674173043600036762, −5.41578218735076917822319617304, −4.20720952097112046725708598268, −3.32030265253004746466443116728, −2.08847827214795889686338457094, −0.75323184643022000560089014451, 0,
0.75323184643022000560089014451, 2.08847827214795889686338457094, 3.32030265253004746466443116728, 4.20720952097112046725708598268, 5.41578218735076917822319617304, 6.27760163357674173043600036762, 6.91768835792201796290064241342, 7.62129537969286697841465014344, 8.601117620805056390712187984724