| L(s) = 1 | − 1.69·2-s − 0.649·3-s + 0.870·4-s − 0.0469·5-s + 1.10·6-s + 3.97·7-s + 1.91·8-s − 2.57·9-s + 0.0795·10-s − 5.59·11-s − 0.564·12-s + 4.37·13-s − 6.74·14-s + 0.0304·15-s − 4.98·16-s + 0.0418·17-s + 4.36·18-s + 19-s − 0.0408·20-s − 2.58·21-s + 9.47·22-s + 3.81·23-s − 1.24·24-s − 4.99·25-s − 7.42·26-s + 3.62·27-s + 3.46·28-s + ⋯ |
| L(s) = 1 | − 1.19·2-s − 0.374·3-s + 0.435·4-s − 0.0210·5-s + 0.449·6-s + 1.50·7-s + 0.676·8-s − 0.859·9-s + 0.0251·10-s − 1.68·11-s − 0.163·12-s + 1.21·13-s − 1.80·14-s + 0.00787·15-s − 1.24·16-s + 0.0101·17-s + 1.02·18-s + 0.229·19-s − 0.00913·20-s − 0.563·21-s + 2.01·22-s + 0.795·23-s − 0.253·24-s − 0.999·25-s − 1.45·26-s + 0.697·27-s + 0.654·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 19 | \( 1 - T \) |
| 211 | \( 1 - T \) |
| good | 2 | \( 1 + 1.69T + 2T^{2} \) |
| 3 | \( 1 + 0.649T + 3T^{2} \) |
| 5 | \( 1 + 0.0469T + 5T^{2} \) |
| 7 | \( 1 - 3.97T + 7T^{2} \) |
| 11 | \( 1 + 5.59T + 11T^{2} \) |
| 13 | \( 1 - 4.37T + 13T^{2} \) |
| 17 | \( 1 - 0.0418T + 17T^{2} \) |
| 23 | \( 1 - 3.81T + 23T^{2} \) |
| 29 | \( 1 - 2.20T + 29T^{2} \) |
| 31 | \( 1 + 1.50T + 31T^{2} \) |
| 37 | \( 1 - 2.23T + 37T^{2} \) |
| 41 | \( 1 - 1.18T + 41T^{2} \) |
| 43 | \( 1 + 7.41T + 43T^{2} \) |
| 47 | \( 1 + 9.98T + 47T^{2} \) |
| 53 | \( 1 + 11.3T + 53T^{2} \) |
| 59 | \( 1 - 10.0T + 59T^{2} \) |
| 61 | \( 1 + 5.26T + 61T^{2} \) |
| 67 | \( 1 + 0.395T + 67T^{2} \) |
| 71 | \( 1 - 2.82T + 71T^{2} \) |
| 73 | \( 1 - 3.30T + 73T^{2} \) |
| 79 | \( 1 + 1.69T + 79T^{2} \) |
| 83 | \( 1 + 12.4T + 83T^{2} \) |
| 89 | \( 1 + 3.52T + 89T^{2} \) |
| 97 | \( 1 - 12.5T + 97T^{2} \) |
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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
−8.143014558646953191051649085230, −7.84262631278580596508770655966, −6.83801397581242103754032170779, −5.77169816457409856037199907078, −5.13670983607180345394336350243, −4.53924224736410016904071836765, −3.20623876626723656459812484069, −2.09862119140301261015354516616, −1.19475010695078952639787805325, 0,
1.19475010695078952639787805325, 2.09862119140301261015354516616, 3.20623876626723656459812484069, 4.53924224736410016904071836765, 5.13670983607180345394336350243, 5.77169816457409856037199907078, 6.83801397581242103754032170779, 7.84262631278580596508770655966, 8.143014558646953191051649085230
