L(s) = 1 | + 0.0526·2-s + 1.28·3-s − 1.99·4-s + 0.451·5-s + 0.0674·6-s − 0.210·8-s − 1.35·9-s + 0.0237·10-s − 4.14·11-s − 2.56·12-s − 0.988·13-s + 0.579·15-s + 3.98·16-s − 17-s − 0.0713·18-s − 2.62·19-s − 0.902·20-s − 0.218·22-s − 1.66·23-s − 0.269·24-s − 4.79·25-s − 0.0520·26-s − 5.58·27-s + 3.20·29-s + 0.0304·30-s − 8.31·31-s + 0.630·32-s + ⋯ |
L(s) = 1 | + 0.0372·2-s + 0.740·3-s − 0.998·4-s + 0.202·5-s + 0.0275·6-s − 0.0743·8-s − 0.452·9-s + 0.00751·10-s − 1.25·11-s − 0.739·12-s − 0.274·13-s + 0.149·15-s + 0.995·16-s − 0.242·17-s − 0.0168·18-s − 0.601·19-s − 0.201·20-s − 0.0465·22-s − 0.348·23-s − 0.0550·24-s − 0.959·25-s − 0.0102·26-s − 1.07·27-s + 0.594·29-s + 0.00556·30-s − 1.49·31-s + 0.111·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 17 | \( 1 + T \) |
good | 2 | \( 1 - 0.0526T + 2T^{2} \) |
| 3 | \( 1 - 1.28T + 3T^{2} \) |
| 5 | \( 1 - 0.451T + 5T^{2} \) |
| 11 | \( 1 + 4.14T + 11T^{2} \) |
| 13 | \( 1 + 0.988T + 13T^{2} \) |
| 19 | \( 1 + 2.62T + 19T^{2} \) |
| 23 | \( 1 + 1.66T + 23T^{2} \) |
| 29 | \( 1 - 3.20T + 29T^{2} \) |
| 31 | \( 1 + 8.31T + 31T^{2} \) |
| 37 | \( 1 - 5.35T + 37T^{2} \) |
| 41 | \( 1 - 3.74T + 41T^{2} \) |
| 43 | \( 1 - 0.271T + 43T^{2} \) |
| 47 | \( 1 + 1.87T + 47T^{2} \) |
| 53 | \( 1 + 4.61T + 53T^{2} \) |
| 59 | \( 1 + 11.1T + 59T^{2} \) |
| 61 | \( 1 + 6.31T + 61T^{2} \) |
| 67 | \( 1 - 11.0T + 67T^{2} \) |
| 71 | \( 1 - 10.4T + 71T^{2} \) |
| 73 | \( 1 + 6.74T + 73T^{2} \) |
| 79 | \( 1 + 0.0703T + 79T^{2} \) |
| 83 | \( 1 + 3.62T + 83T^{2} \) |
| 89 | \( 1 + 6.43T + 89T^{2} \) |
| 97 | \( 1 - 3.59T + 97T^{2} \) |
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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
−9.605936959920023987787543180566, −8.980131206066987550933058776577, −8.101870280408630182831540916125, −7.64689041045680137545626467881, −6.09192461173606283117151727288, −5.28911277903371977442420063419, −4.30038278099463887052981607082, −3.22512401904345068789426993280, −2.16748364347794639154418149068, 0,
2.16748364347794639154418149068, 3.22512401904345068789426993280, 4.30038278099463887052981607082, 5.28911277903371977442420063419, 6.09192461173606283117151727288, 7.64689041045680137545626467881, 8.101870280408630182831540916125, 8.980131206066987550933058776577, 9.605936959920023987787543180566