| L(s) = 1 | − 0.240·2-s − 3-s − 1.94·4-s − 0.991·5-s + 0.240·6-s + 2.53·7-s + 0.946·8-s + 9-s + 0.237·10-s + 0.968·11-s + 1.94·12-s + 2.48·13-s − 0.608·14-s + 0.991·15-s + 3.65·16-s − 17-s − 0.240·18-s − 5.03·19-s + 1.92·20-s − 2.53·21-s − 0.232·22-s − 1.80·23-s − 0.946·24-s − 4.01·25-s − 0.597·26-s − 27-s − 4.92·28-s + ⋯ |
| L(s) = 1 | − 0.169·2-s − 0.577·3-s − 0.971·4-s − 0.443·5-s + 0.0979·6-s + 0.958·7-s + 0.334·8-s + 0.333·9-s + 0.0752·10-s + 0.291·11-s + 0.560·12-s + 0.690·13-s − 0.162·14-s + 0.255·15-s + 0.914·16-s − 0.242·17-s − 0.0565·18-s − 1.15·19-s + 0.430·20-s − 0.553·21-s − 0.0495·22-s − 0.376·23-s − 0.193·24-s − 0.803·25-s − 0.117·26-s − 0.192·27-s − 0.930·28-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 | \( 1 + 0.240T + 2T^{2} \) |
| 5 | \( 1 + 0.991T + 5T^{2} \) |
| 7 | \( 1 - 2.53T + 7T^{2} \) |
| 11 | \( 1 - 0.968T + 11T^{2} \) |
| 13 | \( 1 - 2.48T + 13T^{2} \) |
| 19 | \( 1 + 5.03T + 19T^{2} \) |
| 23 | \( 1 + 1.80T + 23T^{2} \) |
| 29 | \( 1 - 5.21T + 29T^{2} \) |
| 31 | \( 1 - 6.87T + 31T^{2} \) |
| 37 | \( 1 - 2.49T + 37T^{2} \) |
| 41 | \( 1 + 1.02T + 41T^{2} \) |
| 43 | \( 1 + 9.96T + 43T^{2} \) |
| 47 | \( 1 + 3.98T + 47T^{2} \) |
| 53 | \( 1 + 5.76T + 53T^{2} \) |
| 59 | \( 1 - 0.764T + 59T^{2} \) |
| 61 | \( 1 + 6.15T + 61T^{2} \) |
| 67 | \( 1 - 0.538T + 67T^{2} \) |
| 71 | \( 1 - 9.03T + 71T^{2} \) |
| 73 | \( 1 + 2.45T + 73T^{2} \) |
| 79 | \( 1 - 7.60T + 79T^{2} \) |
| 83 | \( 1 - 2.86T + 83T^{2} \) |
| 89 | \( 1 + 0.260T + 89T^{2} \) |
| 97 | \( 1 - 11.8T + 97T^{2} \) |
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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
−7.81141158391993484660569200128, −6.62367275053127990566852033022, −6.19294239510622006545615136461, −5.21724081109937241260330119094, −4.59453066322395359120902210013, −4.18570456283033091978299325801, −3.33732011406215158171730712065, −1.94877155907505196545291466954, −1.08617859617998330349342599052, 0,
1.08617859617998330349342599052, 1.94877155907505196545291466954, 3.33732011406215158171730712065, 4.18570456283033091978299325801, 4.59453066322395359120902210013, 5.21724081109937241260330119094, 6.19294239510622006545615136461, 6.62367275053127990566852033022, 7.81141158391993484660569200128
