| L(s) = 1 | + 1.66·2-s − 3-s + 0.781·4-s + 0.813·5-s − 1.66·6-s + 3.35·7-s − 2.03·8-s + 9-s + 1.35·10-s − 2.37·11-s − 0.781·12-s + 2.35·13-s + 5.59·14-s − 0.813·15-s − 4.95·16-s + 17-s + 1.66·18-s − 3.42·19-s + 0.635·20-s − 3.35·21-s − 3.96·22-s − 1.22·23-s + 2.03·24-s − 4.33·25-s + 3.92·26-s − 27-s + 2.62·28-s + ⋯ |
| L(s) = 1 | + 1.17·2-s − 0.577·3-s + 0.390·4-s + 0.363·5-s − 0.680·6-s + 1.26·7-s − 0.718·8-s + 0.333·9-s + 0.429·10-s − 0.716·11-s − 0.225·12-s + 0.652·13-s + 1.49·14-s − 0.210·15-s − 1.23·16-s + 0.242·17-s + 0.393·18-s − 0.784·19-s + 0.142·20-s − 0.732·21-s − 0.845·22-s − 0.255·23-s + 0.414·24-s − 0.867·25-s + 0.769·26-s − 0.192·27-s + 0.495·28-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 | \( 1 - 1.66T + 2T^{2} \) |
| 5 | \( 1 - 0.813T + 5T^{2} \) |
| 7 | \( 1 - 3.35T + 7T^{2} \) |
| 11 | \( 1 + 2.37T + 11T^{2} \) |
| 13 | \( 1 - 2.35T + 13T^{2} \) |
| 19 | \( 1 + 3.42T + 19T^{2} \) |
| 23 | \( 1 + 1.22T + 23T^{2} \) |
| 29 | \( 1 + 0.420T + 29T^{2} \) |
| 31 | \( 1 + 8.78T + 31T^{2} \) |
| 37 | \( 1 + 3.57T + 37T^{2} \) |
| 41 | \( 1 - 11.2T + 41T^{2} \) |
| 43 | \( 1 - 9.78T + 43T^{2} \) |
| 47 | \( 1 + 2.95T + 47T^{2} \) |
| 53 | \( 1 + 13.7T + 53T^{2} \) |
| 59 | \( 1 + 10.1T + 59T^{2} \) |
| 61 | \( 1 + 10.3T + 61T^{2} \) |
| 67 | \( 1 + 7.75T + 67T^{2} \) |
| 71 | \( 1 + 9.62T + 71T^{2} \) |
| 73 | \( 1 + 4.31T + 73T^{2} \) |
| 79 | \( 1 - 9.31T + 79T^{2} \) |
| 83 | \( 1 - 15.2T + 83T^{2} \) |
| 89 | \( 1 - 5.31T + 89T^{2} \) |
| 97 | \( 1 + 11.5T + 97T^{2} \) |
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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
−7.59724242668648324975228865275, −6.33183760115352579002626409350, −5.93123854252192176785544269271, −5.36913930198330461057427185955, −4.67650277730250925682183445532, −4.19472989263415462893609105621, −3.32441910337726398115217466872, −2.27014917282942637458523099083, −1.51243495430594233950104053933, 0,
1.51243495430594233950104053933, 2.27014917282942637458523099083, 3.32441910337726398115217466872, 4.19472989263415462893609105621, 4.67650277730250925682183445532, 5.36913930198330461057427185955, 5.93123854252192176785544269271, 6.33183760115352579002626409350, 7.59724242668648324975228865275
