| L(s) = 1 | + 2-s − 0.924·3-s + 4-s − 5-s − 0.924·6-s + 8-s − 2.14·9-s − 10-s − 2.83·11-s − 0.924·12-s − 6.76·13-s + 0.924·15-s + 16-s − 17-s − 2.14·18-s + 0.279·19-s − 20-s − 2.83·22-s + 1.28·23-s − 0.924·24-s + 25-s − 6.76·26-s + 4.75·27-s − 8.50·29-s + 0.924·30-s + 3.68·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.533·3-s + 0.5·4-s − 0.447·5-s − 0.377·6-s + 0.353·8-s − 0.715·9-s − 0.316·10-s − 0.853·11-s − 0.266·12-s − 1.87·13-s + 0.238·15-s + 0.250·16-s − 0.242·17-s − 0.505·18-s + 0.0641·19-s − 0.223·20-s − 0.603·22-s + 0.267·23-s − 0.188·24-s + 0.200·25-s − 1.32·26-s + 0.915·27-s − 1.57·29-s + 0.168·30-s + 0.661·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.068985862\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.068985862\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 + 0.924T + 3T^{2} \) |
| 11 | \( 1 + 2.83T + 11T^{2} \) |
| 13 | \( 1 + 6.76T + 13T^{2} \) |
| 19 | \( 1 - 0.279T + 19T^{2} \) |
| 23 | \( 1 - 1.28T + 23T^{2} \) |
| 29 | \( 1 + 8.50T + 29T^{2} \) |
| 31 | \( 1 - 3.68T + 31T^{2} \) |
| 37 | \( 1 - 6.85T + 37T^{2} \) |
| 41 | \( 1 + 6.75T + 41T^{2} \) |
| 43 | \( 1 - 11.2T + 43T^{2} \) |
| 47 | \( 1 + 10.6T + 47T^{2} \) |
| 53 | \( 1 + 13.7T + 53T^{2} \) |
| 59 | \( 1 + 2.60T + 59T^{2} \) |
| 61 | \( 1 + 13.9T + 61T^{2} \) |
| 67 | \( 1 - 11.3T + 67T^{2} \) |
| 71 | \( 1 - 11.0T + 71T^{2} \) |
| 73 | \( 1 - 1.76T + 73T^{2} \) |
| 79 | \( 1 - 11.6T + 79T^{2} \) |
| 83 | \( 1 - 12.9T + 83T^{2} \) |
| 89 | \( 1 + 6.63T + 89T^{2} \) |
| 97 | \( 1 - 0.833T + 97T^{2} \) |
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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
−7.82687615563548547901127514010, −7.02788314966998871328120927148, −6.32929091665668783126283145698, −5.58443186784774758886831360265, −4.91141390235706575080580088282, −4.61571518502004596965253112747, −3.43618312887914129782120783435, −2.76407151561025902228307628228, −2.04415598732604215976502943514, −0.43516997035052387096226219929,
0.43516997035052387096226219929, 2.04415598732604215976502943514, 2.76407151561025902228307628228, 3.43618312887914129782120783435, 4.61571518502004596965253112747, 4.91141390235706575080580088282, 5.58443186784774758886831360265, 6.32929091665668783126283145698, 7.02788314966998871328120927148, 7.82687615563548547901127514010
