| L(s) = 1 | + 0.0687·2-s − 3-s − 1.99·4-s + 0.966·5-s − 0.0687·6-s + 0.824·7-s − 0.274·8-s + 9-s + 0.0664·10-s + 2.54·11-s + 1.99·12-s − 0.576·13-s + 0.0567·14-s − 0.966·15-s + 3.97·16-s − 17-s + 0.0687·18-s + 3.08·19-s − 1.92·20-s − 0.824·21-s + 0.175·22-s − 2.04·23-s + 0.274·24-s − 4.06·25-s − 0.0396·26-s − 27-s − 1.64·28-s + ⋯ |
| L(s) = 1 | + 0.0486·2-s − 0.577·3-s − 0.997·4-s + 0.432·5-s − 0.0280·6-s + 0.311·7-s − 0.0971·8-s + 0.333·9-s + 0.0210·10-s + 0.768·11-s + 0.575·12-s − 0.159·13-s + 0.0151·14-s − 0.249·15-s + 0.992·16-s − 0.242·17-s + 0.0162·18-s + 0.708·19-s − 0.431·20-s − 0.180·21-s + 0.0373·22-s − 0.426·23-s + 0.0560·24-s − 0.813·25-s − 0.00777·26-s − 0.192·27-s − 0.311·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 157 | \( 1 + T \) |
| good | 2 | \( 1 - 0.0687T + 2T^{2} \) |
| 5 | \( 1 - 0.966T + 5T^{2} \) |
| 7 | \( 1 - 0.824T + 7T^{2} \) |
| 11 | \( 1 - 2.54T + 11T^{2} \) |
| 13 | \( 1 + 0.576T + 13T^{2} \) |
| 19 | \( 1 - 3.08T + 19T^{2} \) |
| 23 | \( 1 + 2.04T + 23T^{2} \) |
| 29 | \( 1 - 2.52T + 29T^{2} \) |
| 31 | \( 1 + 6.07T + 31T^{2} \) |
| 37 | \( 1 + 5.31T + 37T^{2} \) |
| 41 | \( 1 - 3.31T + 41T^{2} \) |
| 43 | \( 1 - 4.06T + 43T^{2} \) |
| 47 | \( 1 + 6.63T + 47T^{2} \) |
| 53 | \( 1 + 10.1T + 53T^{2} \) |
| 59 | \( 1 + 10.1T + 59T^{2} \) |
| 61 | \( 1 - 3.84T + 61T^{2} \) |
| 67 | \( 1 - 12.8T + 67T^{2} \) |
| 71 | \( 1 - 6.40T + 71T^{2} \) |
| 73 | \( 1 - 10.2T + 73T^{2} \) |
| 79 | \( 1 + 2.90T + 79T^{2} \) |
| 83 | \( 1 + 3.83T + 83T^{2} \) |
| 89 | \( 1 - 16.7T + 89T^{2} \) |
| 97 | \( 1 + 4.28T + 97T^{2} \) |
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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
−7.58276577009466200733117826851, −6.62422440723860624065516193809, −6.04833334283629732802790072572, −5.27676047266088618704141195522, −4.82517791158823751988677973671, −3.97868574549677702554057999204, −3.36133069107355080946185465272, −2.00979651830204386436476203770, −1.17258655137140479292782355871, 0,
1.17258655137140479292782355871, 2.00979651830204386436476203770, 3.36133069107355080946185465272, 3.97868574549677702554057999204, 4.82517791158823751988677973671, 5.27676047266088618704141195522, 6.04833334283629732802790072572, 6.62422440723860624065516193809, 7.58276577009466200733117826851
