L(s) = 1 | + 1.84·2-s + 1.04·3-s + 1.40·4-s − 1.31·5-s + 1.92·6-s − 2.42·7-s − 1.09·8-s − 1.91·9-s − 2.42·10-s + 11-s + 1.46·12-s − 4.97·13-s − 4.46·14-s − 1.36·15-s − 4.83·16-s − 0.503·17-s − 3.53·18-s − 2.87·19-s − 1.84·20-s − 2.51·21-s + 1.84·22-s + 8.23·23-s − 1.14·24-s − 3.27·25-s − 9.17·26-s − 5.11·27-s − 3.40·28-s + ⋯ |
L(s) = 1 | + 1.30·2-s + 0.601·3-s + 0.702·4-s − 0.587·5-s + 0.784·6-s − 0.914·7-s − 0.387·8-s − 0.638·9-s − 0.766·10-s + 0.301·11-s + 0.422·12-s − 1.37·13-s − 1.19·14-s − 0.353·15-s − 1.20·16-s − 0.122·17-s − 0.833·18-s − 0.660·19-s − 0.412·20-s − 0.549·21-s + 0.393·22-s + 1.71·23-s − 0.233·24-s − 0.654·25-s − 1.79·26-s − 0.984·27-s − 0.642·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 - T \) |
| 131 | \( 1 - T \) |
good | 2 | \( 1 - 1.84T + 2T^{2} \) |
| 3 | \( 1 - 1.04T + 3T^{2} \) |
| 5 | \( 1 + 1.31T + 5T^{2} \) |
| 7 | \( 1 + 2.42T + 7T^{2} \) |
| 13 | \( 1 + 4.97T + 13T^{2} \) |
| 17 | \( 1 + 0.503T + 17T^{2} \) |
| 19 | \( 1 + 2.87T + 19T^{2} \) |
| 23 | \( 1 - 8.23T + 23T^{2} \) |
| 29 | \( 1 + 10.5T + 29T^{2} \) |
| 31 | \( 1 - 9.72T + 31T^{2} \) |
| 37 | \( 1 - 8.51T + 37T^{2} \) |
| 41 | \( 1 - 4.42T + 41T^{2} \) |
| 43 | \( 1 - 0.944T + 43T^{2} \) |
| 47 | \( 1 + 2.77T + 47T^{2} \) |
| 53 | \( 1 + 7.00T + 53T^{2} \) |
| 59 | \( 1 - 7.74T + 59T^{2} \) |
| 61 | \( 1 + 10.5T + 61T^{2} \) |
| 67 | \( 1 - 10.4T + 67T^{2} \) |
| 71 | \( 1 - 2.74T + 71T^{2} \) |
| 73 | \( 1 + 9.98T + 73T^{2} \) |
| 79 | \( 1 + 15.2T + 79T^{2} \) |
| 83 | \( 1 - 14.1T + 83T^{2} \) |
| 89 | \( 1 + 3.34T + 89T^{2} \) |
| 97 | \( 1 + 5.58T + 97T^{2} \) |
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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
−9.323859477534018330219407776213, −8.256009265561775117754696375542, −7.36378234454335290988345458171, −6.51709300646716012335416152971, −5.71148587020054599576526630399, −4.73248946494326560124991508507, −3.95017722311162145827179315418, −3.06833315040968195926688700266, −2.48569198815890300662191229742, 0,
2.48569198815890300662191229742, 3.06833315040968195926688700266, 3.95017722311162145827179315418, 4.73248946494326560124991508507, 5.71148587020054599576526630399, 6.51709300646716012335416152971, 7.36378234454335290988345458171, 8.256009265561775117754696375542, 9.323859477534018330219407776213