L(s) = 1 | + 1.70·2-s + 3-s + 0.895·4-s − 0.611·5-s + 1.70·6-s − 2.23·7-s − 1.87·8-s + 9-s − 1.04·10-s + 3.57·11-s + 0.895·12-s − 0.914·13-s − 3.79·14-s − 0.611·15-s − 4.98·16-s + 17-s + 1.70·18-s − 1.88·19-s − 0.547·20-s − 2.23·21-s + 6.08·22-s + 0.756·23-s − 1.87·24-s − 4.62·25-s − 1.55·26-s + 27-s − 1.99·28-s + ⋯ |
L(s) = 1 | + 1.20·2-s + 0.577·3-s + 0.447·4-s − 0.273·5-s + 0.694·6-s − 0.843·7-s − 0.664·8-s + 0.333·9-s − 0.329·10-s + 1.07·11-s + 0.258·12-s − 0.253·13-s − 1.01·14-s − 0.157·15-s − 1.24·16-s + 0.242·17-s + 0.401·18-s − 0.432·19-s − 0.122·20-s − 0.487·21-s + 1.29·22-s + 0.157·23-s − 0.383·24-s − 0.925·25-s − 0.305·26-s + 0.192·27-s − 0.377·28-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 | \( 1 - 1.70T + 2T^{2} \) |
| 5 | \( 1 + 0.611T + 5T^{2} \) |
| 7 | \( 1 + 2.23T + 7T^{2} \) |
| 11 | \( 1 - 3.57T + 11T^{2} \) |
| 13 | \( 1 + 0.914T + 13T^{2} \) |
| 19 | \( 1 + 1.88T + 19T^{2} \) |
| 23 | \( 1 - 0.756T + 23T^{2} \) |
| 29 | \( 1 - 6.86T + 29T^{2} \) |
| 31 | \( 1 + 0.896T + 31T^{2} \) |
| 37 | \( 1 - 2.98T + 37T^{2} \) |
| 41 | \( 1 + 11.2T + 41T^{2} \) |
| 43 | \( 1 - 4.43T + 43T^{2} \) |
| 47 | \( 1 - 3.37T + 47T^{2} \) |
| 53 | \( 1 - 0.194T + 53T^{2} \) |
| 59 | \( 1 + 7.41T + 59T^{2} \) |
| 61 | \( 1 + 11.2T + 61T^{2} \) |
| 67 | \( 1 + 3.30T + 67T^{2} \) |
| 71 | \( 1 + 10.8T + 71T^{2} \) |
| 73 | \( 1 + 6.08T + 73T^{2} \) |
| 79 | \( 1 + 5.81T + 79T^{2} \) |
| 83 | \( 1 - 7.26T + 83T^{2} \) |
| 89 | \( 1 - 1.69T + 89T^{2} \) |
| 97 | \( 1 + 1.94T + 97T^{2} \) |
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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
−7.31637004351026915937229343743, −6.51642683116011119630443612556, −6.21042716648836401103188858552, −5.28629579413244528740860205918, −4.39182130208670935333106653846, −3.99738985323571872734798645085, −3.21115572941452943241205752777, −2.71118448062980566442182083316, −1.50843320634017699011597218634, 0,
1.50843320634017699011597218634, 2.71118448062980566442182083316, 3.21115572941452943241205752777, 3.99738985323571872734798645085, 4.39182130208670935333106653846, 5.28629579413244528740860205918, 6.21042716648836401103188858552, 6.51642683116011119630443612556, 7.31637004351026915937229343743