| L(s) = 1 | + 1.34·2-s − 3-s − 0.178·4-s + 4.12·5-s − 1.34·6-s − 7-s − 2.93·8-s + 9-s + 5.57·10-s − 1.10·11-s + 0.178·12-s − 0.242·13-s − 1.34·14-s − 4.12·15-s − 3.61·16-s − 6.86·17-s + 1.34·18-s − 6.00·19-s − 0.735·20-s + 21-s − 1.49·22-s + 6.68·23-s + 2.93·24-s + 12.0·25-s − 0.326·26-s − 27-s + 0.178·28-s + ⋯ |
| L(s) = 1 | + 0.954·2-s − 0.577·3-s − 0.0890·4-s + 1.84·5-s − 0.551·6-s − 0.377·7-s − 1.03·8-s + 0.333·9-s + 1.76·10-s − 0.333·11-s + 0.0514·12-s − 0.0671·13-s − 0.360·14-s − 1.06·15-s − 0.903·16-s − 1.66·17-s + 0.318·18-s − 1.37·19-s − 0.164·20-s + 0.218·21-s − 0.317·22-s + 1.39·23-s + 0.600·24-s + 2.41·25-s − 0.0641·26-s − 0.192·27-s + 0.0336·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 + T \) |
| good | 2 | \( 1 - 1.34T + 2T^{2} \) |
| 5 | \( 1 - 4.12T + 5T^{2} \) |
| 11 | \( 1 + 1.10T + 11T^{2} \) |
| 13 | \( 1 + 0.242T + 13T^{2} \) |
| 17 | \( 1 + 6.86T + 17T^{2} \) |
| 19 | \( 1 + 6.00T + 19T^{2} \) |
| 23 | \( 1 - 6.68T + 23T^{2} \) |
| 29 | \( 1 - 5.89T + 29T^{2} \) |
| 31 | \( 1 - 3.25T + 31T^{2} \) |
| 37 | \( 1 + 4.50T + 37T^{2} \) |
| 41 | \( 1 - 0.728T + 41T^{2} \) |
| 43 | \( 1 - 10.5T + 43T^{2} \) |
| 47 | \( 1 + 3.21T + 47T^{2} \) |
| 53 | \( 1 + 10.2T + 53T^{2} \) |
| 59 | \( 1 + 0.167T + 59T^{2} \) |
| 61 | \( 1 + 3.78T + 61T^{2} \) |
| 67 | \( 1 + 3.31T + 67T^{2} \) |
| 71 | \( 1 - 1.98T + 71T^{2} \) |
| 73 | \( 1 + 12.9T + 73T^{2} \) |
| 79 | \( 1 + 15.3T + 79T^{2} \) |
| 83 | \( 1 - 1.16T + 83T^{2} \) |
| 89 | \( 1 + 17.3T + 89T^{2} \) |
| 97 | \( 1 + 16.0T + 97T^{2} \) |
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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
−6.77497905986623344221891037449, −6.69767461142203620484256402157, −5.94485666066694213338629795563, −5.46689418595366382130865055630, −4.63645743130153312970442687868, −4.35079636492711564071671832481, −2.88506359820519249051529417262, −2.52800357875086573321887216268, −1.42146746839723125168310242119, 0,
1.42146746839723125168310242119, 2.52800357875086573321887216268, 2.88506359820519249051529417262, 4.35079636492711564071671832481, 4.63645743130153312970442687868, 5.46689418595366382130865055630, 5.94485666066694213338629795563, 6.69767461142203620484256402157, 6.77497905986623344221891037449
