| L(s) = 1 | + 0.556·2-s + 3-s − 1.68·4-s + 0.267·5-s + 0.556·6-s − 0.618·7-s − 2.05·8-s + 9-s + 0.149·10-s − 5.42·11-s − 1.68·12-s − 0.754·13-s − 0.344·14-s + 0.267·15-s + 2.23·16-s + 17-s + 0.556·18-s + 7.96·19-s − 0.452·20-s − 0.618·21-s − 3.02·22-s + 7.44·23-s − 2.05·24-s − 4.92·25-s − 0.419·26-s + 27-s + 1.04·28-s + ⋯ |
| L(s) = 1 | + 0.393·2-s + 0.577·3-s − 0.844·4-s + 0.119·5-s + 0.227·6-s − 0.233·7-s − 0.726·8-s + 0.333·9-s + 0.0471·10-s − 1.63·11-s − 0.487·12-s − 0.209·13-s − 0.0920·14-s + 0.0691·15-s + 0.558·16-s + 0.242·17-s + 0.131·18-s + 1.82·19-s − 0.101·20-s − 0.135·21-s − 0.643·22-s + 1.55·23-s − 0.419·24-s − 0.985·25-s − 0.0823·26-s + 0.192·27-s + 0.197·28-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 | \( 1 - 0.556T + 2T^{2} \) |
| 5 | \( 1 - 0.267T + 5T^{2} \) |
| 7 | \( 1 + 0.618T + 7T^{2} \) |
| 11 | \( 1 + 5.42T + 11T^{2} \) |
| 13 | \( 1 + 0.754T + 13T^{2} \) |
| 19 | \( 1 - 7.96T + 19T^{2} \) |
| 23 | \( 1 - 7.44T + 23T^{2} \) |
| 29 | \( 1 - 5.06T + 29T^{2} \) |
| 31 | \( 1 + 10.4T + 31T^{2} \) |
| 37 | \( 1 + 5.93T + 37T^{2} \) |
| 41 | \( 1 - 2.79T + 41T^{2} \) |
| 43 | \( 1 - 10.1T + 43T^{2} \) |
| 47 | \( 1 - 1.11T + 47T^{2} \) |
| 53 | \( 1 + 4.46T + 53T^{2} \) |
| 59 | \( 1 + 9.54T + 59T^{2} \) |
| 61 | \( 1 - 5.90T + 61T^{2} \) |
| 67 | \( 1 - 0.814T + 67T^{2} \) |
| 71 | \( 1 + 2.12T + 71T^{2} \) |
| 73 | \( 1 + 0.654T + 73T^{2} \) |
| 79 | \( 1 + 4.90T + 79T^{2} \) |
| 83 | \( 1 - 7.55T + 83T^{2} \) |
| 89 | \( 1 + 2.20T + 89T^{2} \) |
| 97 | \( 1 + 1.22T + 97T^{2} \) |
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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
−7.69663402428089390780082719506, −6.94664557520431719425909395002, −5.77962599396669170680492591954, −5.28480705125146461108647989708, −4.82527841872033301255236151803, −3.77924620644657598230844257541, −3.12902930973203536401637392081, −2.58578654146080735018067828757, −1.23820246947135476480038276876, 0,
1.23820246947135476480038276876, 2.58578654146080735018067828757, 3.12902930973203536401637392081, 3.77924620644657598230844257541, 4.82527841872033301255236151803, 5.28480705125146461108647989708, 5.77962599396669170680492591954, 6.94664557520431719425909395002, 7.69663402428089390780082719506
