| L(s) = 1 | + 2-s + 0.0458·3-s + 4-s − 0.293·5-s + 0.0458·6-s + 1.15·7-s + 8-s − 2.99·9-s − 0.293·10-s + 0.240·11-s + 0.0458·12-s − 4.13·13-s + 1.15·14-s − 0.0134·15-s + 16-s − 0.943·17-s − 2.99·18-s + 6.60·19-s − 0.293·20-s + 0.0530·21-s + 0.240·22-s + 0.399·23-s + 0.0458·24-s − 4.91·25-s − 4.13·26-s − 0.274·27-s + 1.15·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.0264·3-s + 0.5·4-s − 0.131·5-s + 0.0187·6-s + 0.437·7-s + 0.353·8-s − 0.999·9-s − 0.0929·10-s + 0.0724·11-s + 0.0132·12-s − 1.14·13-s + 0.309·14-s − 0.00347·15-s + 0.250·16-s − 0.228·17-s − 0.706·18-s + 1.51·19-s − 0.0656·20-s + 0.0115·21-s + 0.0512·22-s + 0.0833·23-s + 0.00935·24-s − 0.982·25-s − 0.811·26-s − 0.0528·27-s + 0.218·28-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 | \( 1 - 0.0458T + 3T^{2} \) |
| 5 | \( 1 + 0.293T + 5T^{2} \) |
| 7 | \( 1 - 1.15T + 7T^{2} \) |
| 11 | \( 1 - 0.240T + 11T^{2} \) |
| 13 | \( 1 + 4.13T + 13T^{2} \) |
| 17 | \( 1 + 0.943T + 17T^{2} \) |
| 19 | \( 1 - 6.60T + 19T^{2} \) |
| 23 | \( 1 - 0.399T + 23T^{2} \) |
| 29 | \( 1 + 3.17T + 29T^{2} \) |
| 31 | \( 1 + 7.17T + 31T^{2} \) |
| 37 | \( 1 - 5.32T + 37T^{2} \) |
| 41 | \( 1 - 4.32T + 41T^{2} \) |
| 43 | \( 1 - 3.21T + 43T^{2} \) |
| 47 | \( 1 + 7.89T + 47T^{2} \) |
| 53 | \( 1 - 3.79T + 53T^{2} \) |
| 59 | \( 1 + 4.27T + 59T^{2} \) |
| 61 | \( 1 - 4.79T + 61T^{2} \) |
| 67 | \( 1 + 13.6T + 67T^{2} \) |
| 71 | \( 1 + 12.5T + 71T^{2} \) |
| 73 | \( 1 + 7.96T + 73T^{2} \) |
| 79 | \( 1 + 1.64T + 79T^{2} \) |
| 83 | \( 1 - 3.91T + 83T^{2} \) |
| 89 | \( 1 + 3.39T + 89T^{2} \) |
| 97 | \( 1 - 1.41T + 97T^{2} \) |
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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
−7.58033895013805421284793192901, −7.16224560094141104109423675004, −6.03871720885389453558363656406, −5.54119000185580111473465672776, −4.90399131798765883747940831550, −4.09739213745283877470638578821, −3.19701492264063534938027517199, −2.53906017561742739725572761999, −1.54411388982279829095426199383, 0,
1.54411388982279829095426199383, 2.53906017561742739725572761999, 3.19701492264063534938027517199, 4.09739213745283877470638578821, 4.90399131798765883747940831550, 5.54119000185580111473465672776, 6.03871720885389453558363656406, 7.16224560094141104109423675004, 7.58033895013805421284793192901
