| L(s) = 1 | − 1.65·2-s + 1.21·3-s + 0.751·4-s + 5-s − 2.01·6-s − 7-s + 2.07·8-s − 1.52·9-s − 1.65·10-s − 5.02·11-s + 0.912·12-s − 1.70·13-s + 1.65·14-s + 1.21·15-s − 4.93·16-s + 4.99·17-s + 2.52·18-s − 0.872·19-s + 0.751·20-s − 1.21·21-s + 8.33·22-s − 2.73·23-s + 2.51·24-s + 25-s + 2.81·26-s − 5.49·27-s − 0.751·28-s + ⋯ |
| L(s) = 1 | − 1.17·2-s + 0.701·3-s + 0.375·4-s + 0.447·5-s − 0.822·6-s − 0.377·7-s + 0.732·8-s − 0.507·9-s − 0.524·10-s − 1.51·11-s + 0.263·12-s − 0.471·13-s + 0.443·14-s + 0.313·15-s − 1.23·16-s + 1.21·17-s + 0.595·18-s − 0.200·19-s + 0.167·20-s − 0.265·21-s + 1.77·22-s − 0.569·23-s + 0.513·24-s + 0.200·25-s + 0.553·26-s − 1.05·27-s − 0.141·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 229 | \( 1 - T \) |
| good | 2 | \( 1 + 1.65T + 2T^{2} \) |
| 3 | \( 1 - 1.21T + 3T^{2} \) |
| 11 | \( 1 + 5.02T + 11T^{2} \) |
| 13 | \( 1 + 1.70T + 13T^{2} \) |
| 17 | \( 1 - 4.99T + 17T^{2} \) |
| 19 | \( 1 + 0.872T + 19T^{2} \) |
| 23 | \( 1 + 2.73T + 23T^{2} \) |
| 29 | \( 1 - 0.150T + 29T^{2} \) |
| 31 | \( 1 - 7.20T + 31T^{2} \) |
| 37 | \( 1 - 4.43T + 37T^{2} \) |
| 41 | \( 1 - 6.95T + 41T^{2} \) |
| 43 | \( 1 - 3.64T + 43T^{2} \) |
| 47 | \( 1 + 4.69T + 47T^{2} \) |
| 53 | \( 1 - 5.70T + 53T^{2} \) |
| 59 | \( 1 - 12.3T + 59T^{2} \) |
| 61 | \( 1 + 0.319T + 61T^{2} \) |
| 67 | \( 1 - 1.80T + 67T^{2} \) |
| 71 | \( 1 + 3.63T + 71T^{2} \) |
| 73 | \( 1 + 6.37T + 73T^{2} \) |
| 79 | \( 1 - 6.92T + 79T^{2} \) |
| 83 | \( 1 + 2.79T + 83T^{2} \) |
| 89 | \( 1 + 7.82T + 89T^{2} \) |
| 97 | \( 1 + 7.30T + 97T^{2} \) |
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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
−7.75894742052532462847818801210, −7.21984447539501606371224248326, −6.13850753257703605150584811149, −5.49893761284264970290416526787, −4.73027265016814496009075205962, −3.71776955690330762985200079631, −2.61072609497594210279070109382, −2.41870768061837310643501278439, −1.07799870319071291645807112478, 0,
1.07799870319071291645807112478, 2.41870768061837310643501278439, 2.61072609497594210279070109382, 3.71776955690330762985200079631, 4.73027265016814496009075205962, 5.49893761284264970290416526787, 6.13850753257703605150584811149, 7.21984447539501606371224248326, 7.75894742052532462847818801210
