L(s) = 1 | + 2-s − 1.98·3-s + 4-s + 5-s − 1.98·6-s − 4.22·7-s + 8-s + 0.925·9-s + 10-s + 0.530·11-s − 1.98·12-s − 0.100·13-s − 4.22·14-s − 1.98·15-s + 16-s − 3.01·17-s + 0.925·18-s − 1.24·19-s + 20-s + 8.36·21-s + 0.530·22-s + 8.77·23-s − 1.98·24-s + 25-s − 0.100·26-s + 4.11·27-s − 4.22·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.14·3-s + 0.5·4-s + 0.447·5-s − 0.808·6-s − 1.59·7-s + 0.353·8-s + 0.308·9-s + 0.316·10-s + 0.159·11-s − 0.571·12-s − 0.0279·13-s − 1.12·14-s − 0.511·15-s + 0.250·16-s − 0.732·17-s + 0.218·18-s − 0.285·19-s + 0.223·20-s + 1.82·21-s + 0.113·22-s + 1.83·23-s − 0.404·24-s + 0.200·25-s − 0.0197·26-s + 0.791·27-s − 0.797·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 601 | \( 1 + T \) |
good | 3 | \( 1 + 1.98T + 3T^{2} \) |
| 7 | \( 1 + 4.22T + 7T^{2} \) |
| 11 | \( 1 - 0.530T + 11T^{2} \) |
| 13 | \( 1 + 0.100T + 13T^{2} \) |
| 17 | \( 1 + 3.01T + 17T^{2} \) |
| 19 | \( 1 + 1.24T + 19T^{2} \) |
| 23 | \( 1 - 8.77T + 23T^{2} \) |
| 29 | \( 1 - 1.18T + 29T^{2} \) |
| 31 | \( 1 - 3.73T + 31T^{2} \) |
| 37 | \( 1 - 5.95T + 37T^{2} \) |
| 41 | \( 1 + 10.4T + 41T^{2} \) |
| 43 | \( 1 - 9.82T + 43T^{2} \) |
| 47 | \( 1 + 2.60T + 47T^{2} \) |
| 53 | \( 1 - 6.07T + 53T^{2} \) |
| 59 | \( 1 + 11.8T + 59T^{2} \) |
| 61 | \( 1 + 9.97T + 61T^{2} \) |
| 67 | \( 1 + 7.26T + 67T^{2} \) |
| 71 | \( 1 - 15.6T + 71T^{2} \) |
| 73 | \( 1 + 15.2T + 73T^{2} \) |
| 79 | \( 1 + 8.86T + 79T^{2} \) |
| 83 | \( 1 + 10.8T + 83T^{2} \) |
| 89 | \( 1 + 12.3T + 89T^{2} \) |
| 97 | \( 1 - 9.19T + 97T^{2} \) |
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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
−7.20057231330763375386403839430, −6.75823929668629107045995094926, −6.17974713964334762257300665363, −5.75737969119074214517256830038, −4.89093591167500446091813677454, −4.26551827292511229379281529678, −3.11790891543187616175474258462, −2.66866704741112605995988308556, −1.19298505667726138047473667191, 0,
1.19298505667726138047473667191, 2.66866704741112605995988308556, 3.11790891543187616175474258462, 4.26551827292511229379281529678, 4.89093591167500446091813677454, 5.75737969119074214517256830038, 6.17974713964334762257300665363, 6.75823929668629107045995094926, 7.20057231330763375386403839430