L(s) = 1 | + 2-s + 2.65·3-s + 4-s − 2.01·5-s + 2.65·6-s − 3.04·7-s + 8-s + 4.03·9-s − 2.01·10-s − 0.414·11-s + 2.65·12-s + 0.200·13-s − 3.04·14-s − 5.35·15-s + 16-s − 2.08·17-s + 4.03·18-s − 2.37·19-s − 2.01·20-s − 8.07·21-s − 0.414·22-s + 1.24·23-s + 2.65·24-s − 0.920·25-s + 0.200·26-s + 2.75·27-s − 3.04·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.53·3-s + 0.5·4-s − 0.903·5-s + 1.08·6-s − 1.15·7-s + 0.353·8-s + 1.34·9-s − 0.638·10-s − 0.124·11-s + 0.765·12-s + 0.0556·13-s − 0.813·14-s − 1.38·15-s + 0.250·16-s − 0.505·17-s + 0.951·18-s − 0.544·19-s − 0.451·20-s − 1.76·21-s − 0.0883·22-s + 0.259·23-s + 0.541·24-s − 0.184·25-s + 0.0393·26-s + 0.530·27-s − 0.575·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3001 | \( 1+O(T) \) |
good | 3 | \( 1 - 2.65T + 3T^{2} \) |
| 5 | \( 1 + 2.01T + 5T^{2} \) |
| 7 | \( 1 + 3.04T + 7T^{2} \) |
| 11 | \( 1 + 0.414T + 11T^{2} \) |
| 13 | \( 1 - 0.200T + 13T^{2} \) |
| 17 | \( 1 + 2.08T + 17T^{2} \) |
| 19 | \( 1 + 2.37T + 19T^{2} \) |
| 23 | \( 1 - 1.24T + 23T^{2} \) |
| 29 | \( 1 + 4.35T + 29T^{2} \) |
| 31 | \( 1 - 2.46T + 31T^{2} \) |
| 37 | \( 1 + 1.62T + 37T^{2} \) |
| 41 | \( 1 + 6.87T + 41T^{2} \) |
| 43 | \( 1 + 9.27T + 43T^{2} \) |
| 47 | \( 1 + 7.77T + 47T^{2} \) |
| 53 | \( 1 + 0.0989T + 53T^{2} \) |
| 59 | \( 1 - 4.70T + 59T^{2} \) |
| 61 | \( 1 - 14.6T + 61T^{2} \) |
| 67 | \( 1 - 9.09T + 67T^{2} \) |
| 71 | \( 1 + 6.36T + 71T^{2} \) |
| 73 | \( 1 + 12.1T + 73T^{2} \) |
| 79 | \( 1 + 9.34T + 79T^{2} \) |
| 83 | \( 1 - 10.3T + 83T^{2} \) |
| 89 | \( 1 + 5.45T + 89T^{2} \) |
| 97 | \( 1 + 13.9T + 97T^{2} \) |
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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
−7.79237323719240989993021694609, −6.94734186664294241516793387691, −6.59957086844625303329880116984, −5.49380754629961722578154460731, −4.49864192509634095730937297937, −3.75850601454906675144793307252, −3.37661851910510909543904303404, −2.65837789499398366845978466487, −1.77579129592883001138024986772, 0,
1.77579129592883001138024986772, 2.65837789499398366845978466487, 3.37661851910510909543904303404, 3.75850601454906675144793307252, 4.49864192509634095730937297937, 5.49380754629961722578154460731, 6.59957086844625303329880116984, 6.94734186664294241516793387691, 7.79237323719240989993021694609