L(s) = 1 | + 2-s − 3.37·3-s + 4-s − 4.42·5-s − 3.37·6-s + 1.01·7-s + 8-s + 8.35·9-s − 4.42·10-s + 3.10·11-s − 3.37·12-s − 4.40·13-s + 1.01·14-s + 14.9·15-s + 16-s + 0.837·17-s + 8.35·18-s + 6.78·19-s − 4.42·20-s − 3.41·21-s + 3.10·22-s + 2.91·23-s − 3.37·24-s + 14.5·25-s − 4.40·26-s − 18.0·27-s + 1.01·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.94·3-s + 0.5·4-s − 1.97·5-s − 1.37·6-s + 0.382·7-s + 0.353·8-s + 2.78·9-s − 1.39·10-s + 0.937·11-s − 0.972·12-s − 1.22·13-s + 0.270·14-s + 3.85·15-s + 0.250·16-s + 0.203·17-s + 1.97·18-s + 1.55·19-s − 0.989·20-s − 0.744·21-s + 0.662·22-s + 0.607·23-s − 0.687·24-s + 2.91·25-s − 0.863·26-s − 3.47·27-s + 0.191·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1502 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1502 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 751 | \( 1 - T \) |
good | 3 | \( 1 + 3.37T + 3T^{2} \) |
| 5 | \( 1 + 4.42T + 5T^{2} \) |
| 7 | \( 1 - 1.01T + 7T^{2} \) |
| 11 | \( 1 - 3.10T + 11T^{2} \) |
| 13 | \( 1 + 4.40T + 13T^{2} \) |
| 17 | \( 1 - 0.837T + 17T^{2} \) |
| 19 | \( 1 - 6.78T + 19T^{2} \) |
| 23 | \( 1 - 2.91T + 23T^{2} \) |
| 29 | \( 1 + 6.10T + 29T^{2} \) |
| 31 | \( 1 + 6.14T + 31T^{2} \) |
| 37 | \( 1 + 11.5T + 37T^{2} \) |
| 41 | \( 1 - 6.65T + 41T^{2} \) |
| 43 | \( 1 + 7.74T + 43T^{2} \) |
| 47 | \( 1 + 3.07T + 47T^{2} \) |
| 53 | \( 1 - 11.4T + 53T^{2} \) |
| 59 | \( 1 - 0.695T + 59T^{2} \) |
| 61 | \( 1 - 4.68T + 61T^{2} \) |
| 67 | \( 1 + 4.54T + 67T^{2} \) |
| 71 | \( 1 + 7.73T + 71T^{2} \) |
| 73 | \( 1 + 1.55T + 73T^{2} \) |
| 79 | \( 1 + 2.20T + 79T^{2} \) |
| 83 | \( 1 - 6.21T + 83T^{2} \) |
| 89 | \( 1 + 0.410T + 89T^{2} \) |
| 97 | \( 1 + 0.409T + 97T^{2} \) |
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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
−9.200934256292264774349275238660, −7.73458894964343075584090940464, −7.19645341052473012959156940382, −6.79891352509249359549540018453, −5.38769415965709949050655266253, −5.04111776769349297975708012291, −4.15214027400400402246162605159, −3.47780742911357661011785047431, −1.30933912099619989462661710760, 0,
1.30933912099619989462661710760, 3.47780742911357661011785047431, 4.15214027400400402246162605159, 5.04111776769349297975708012291, 5.38769415965709949050655266253, 6.79891352509249359549540018453, 7.19645341052473012959156940382, 7.73458894964343075584090940464, 9.200934256292264774349275238660