| L(s) = 1 | − 2.41·2-s + 2.90·3-s + 3.85·4-s − 2.04·5-s − 7.04·6-s + 0.763·7-s − 4.49·8-s + 5.46·9-s + 4.95·10-s + 0.863·11-s + 11.2·12-s − 5.11·13-s − 1.84·14-s − 5.96·15-s + 3.15·16-s − 3.20·17-s − 13.2·18-s − 5.40·19-s − 7.89·20-s + 2.22·21-s − 2.08·22-s + 7.89·23-s − 13.0·24-s − 0.804·25-s + 12.3·26-s + 7.18·27-s + 2.94·28-s + ⋯ |
| L(s) = 1 | − 1.71·2-s + 1.68·3-s + 1.92·4-s − 0.916·5-s − 2.87·6-s + 0.288·7-s − 1.58·8-s + 1.82·9-s + 1.56·10-s + 0.260·11-s + 3.23·12-s − 1.41·13-s − 0.493·14-s − 1.53·15-s + 0.789·16-s − 0.777·17-s − 3.11·18-s − 1.23·19-s − 1.76·20-s + 0.485·21-s − 0.445·22-s + 1.64·23-s − 2.66·24-s − 0.160·25-s + 2.42·26-s + 1.38·27-s + 0.556·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 4007 | \( 1+O(T) \) |
| good | 2 | \( 1 + 2.41T + 2T^{2} \) |
| 3 | \( 1 - 2.90T + 3T^{2} \) |
| 5 | \( 1 + 2.04T + 5T^{2} \) |
| 7 | \( 1 - 0.763T + 7T^{2} \) |
| 11 | \( 1 - 0.863T + 11T^{2} \) |
| 13 | \( 1 + 5.11T + 13T^{2} \) |
| 17 | \( 1 + 3.20T + 17T^{2} \) |
| 19 | \( 1 + 5.40T + 19T^{2} \) |
| 23 | \( 1 - 7.89T + 23T^{2} \) |
| 29 | \( 1 + 2.47T + 29T^{2} \) |
| 31 | \( 1 - 8.16T + 31T^{2} \) |
| 37 | \( 1 - 3.95T + 37T^{2} \) |
| 41 | \( 1 - 4.36T + 41T^{2} \) |
| 43 | \( 1 + 3.96T + 43T^{2} \) |
| 47 | \( 1 + 6.69T + 47T^{2} \) |
| 53 | \( 1 + 1.16T + 53T^{2} \) |
| 59 | \( 1 - 14.3T + 59T^{2} \) |
| 61 | \( 1 + 5.56T + 61T^{2} \) |
| 67 | \( 1 - 5.90T + 67T^{2} \) |
| 71 | \( 1 + 12.0T + 71T^{2} \) |
| 73 | \( 1 + 13.5T + 73T^{2} \) |
| 79 | \( 1 + 2.05T + 79T^{2} \) |
| 83 | \( 1 + 5.37T + 83T^{2} \) |
| 89 | \( 1 - 0.285T + 89T^{2} \) |
| 97 | \( 1 - 8.86T + 97T^{2} \) |
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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
−8.366757523326632112833615377880, −7.63831378997663202303412520612, −7.15379760564071790058498656281, −6.52700607303093891149659005275, −4.74792430211014156903244703446, −4.10477510089250186404861899073, −2.89113507530236374913218947979, −2.40998623813914784996637165628, −1.42567173147981851121495581443, 0,
1.42567173147981851121495581443, 2.40998623813914784996637165628, 2.89113507530236374913218947979, 4.10477510089250186404861899073, 4.74792430211014156903244703446, 6.52700607303093891149659005275, 7.15379760564071790058498656281, 7.63831378997663202303412520612, 8.366757523326632112833615377880
