| L(s) = 1 | − 2-s − 0.337·3-s + 4-s − 2.62·5-s + 0.337·6-s − 0.195·7-s − 8-s − 2.88·9-s + 2.62·10-s − 11-s − 0.337·12-s − 2.97·13-s + 0.195·14-s + 0.886·15-s + 16-s + 0.422·17-s + 2.88·18-s − 2.62·20-s + 0.0659·21-s + 22-s − 1.65·23-s + 0.337·24-s + 1.90·25-s + 2.97·26-s + 1.98·27-s − 0.195·28-s + 2.77·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.194·3-s + 0.5·4-s − 1.17·5-s + 0.137·6-s − 0.0739·7-s − 0.353·8-s − 0.962·9-s + 0.831·10-s − 0.301·11-s − 0.0974·12-s − 0.823·13-s + 0.0522·14-s + 0.228·15-s + 0.250·16-s + 0.102·17-s + 0.680·18-s − 0.587·20-s + 0.0143·21-s + 0.213·22-s − 0.344·23-s + 0.0688·24-s + 0.381·25-s + 0.582·26-s + 0.382·27-s − 0.0369·28-s + 0.515·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1893820143\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1893820143\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 0.337T + 3T^{2} \) |
| 5 | \( 1 + 2.62T + 5T^{2} \) |
| 7 | \( 1 + 0.195T + 7T^{2} \) |
| 13 | \( 1 + 2.97T + 13T^{2} \) |
| 17 | \( 1 - 0.422T + 17T^{2} \) |
| 23 | \( 1 + 1.65T + 23T^{2} \) |
| 29 | \( 1 - 2.77T + 29T^{2} \) |
| 31 | \( 1 - 7.95T + 31T^{2} \) |
| 37 | \( 1 + 9.69T + 37T^{2} \) |
| 41 | \( 1 + 6.33T + 41T^{2} \) |
| 43 | \( 1 + 0.285T + 43T^{2} \) |
| 47 | \( 1 - 1.99T + 47T^{2} \) |
| 53 | \( 1 + 3.41T + 53T^{2} \) |
| 59 | \( 1 + 14.2T + 59T^{2} \) |
| 61 | \( 1 + 8.39T + 61T^{2} \) |
| 67 | \( 1 + 6.72T + 67T^{2} \) |
| 71 | \( 1 + 1.72T + 71T^{2} \) |
| 73 | \( 1 + 13.5T + 73T^{2} \) |
| 79 | \( 1 + 7.36T + 79T^{2} \) |
| 83 | \( 1 - 7.03T + 83T^{2} \) |
| 89 | \( 1 + 9.88T + 89T^{2} \) |
| 97 | \( 1 + 10.0T + 97T^{2} \) |
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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
−7.943282574271418983589400788715, −7.33000322513891660009550854874, −6.60795703284053649360090259460, −5.88786923869254171302024534325, −4.99760894438260868947173278728, −4.36236907498680731703264404976, −3.22606082706273761792783878572, −2.83434796628839755020306125368, −1.60799580203192429226256147486, −0.23726028074176982806347919052,
0.23726028074176982806347919052, 1.60799580203192429226256147486, 2.83434796628839755020306125368, 3.22606082706273761792783878572, 4.36236907498680731703264404976, 4.99760894438260868947173278728, 5.88786923869254171302024534325, 6.60795703284053649360090259460, 7.33000322513891660009550854874, 7.943282574271418983589400788715
